MEASURES IN LINEAR TOPOLOGICAL SPACES

CHAPTER IV

M E A S U R E S IN L I N E A R T O P O L O G I C A L S P A C E S

1 . Basic D e f i n i t i o n s 1.1. Cylinder Sets

In this chapter we study measures in Hnear topological spaces. We will restrict ourselves to considering measures in spaces Φ' which are adjoint to some linear topological space Φ, We will first study measures on the simplest sets in Φ'—the cyHnder sets. Following this, measures on sets of a more general form will be considered. L e t us define the notion of a cylinder set in the space Φ'. We choose any fixed elements φι, in Φ. T o each element F ΕΦ' there corresponds the point ((F, ( F , φ^)) in n-dimensional space R^, T h u s the elements 9^1» ···> 9 n in ^ define a mapping F-^{{F,ψ,\.,.,{F,φ^)) of φ' into R^, Let ^ be a given set in als F such that

(1)

and consider the set Ζ of all linear function­

We call this set Ζ the cylinder set defined by the elements φι, and the set A in R^. As examples of cylinder sets we may consider the half-spaces in Φ' defined by an inequality of the form {F, φ) ^ a, and also sets of a more general type—strips, defined by systems of inequalities <(^^,φ.)

l ^ k ^ n .

One can give another definition of a cyHnder set. L e t φι, φ^ be elements of Φ. We decompose the space Φ' into cosets, regarding all linear functionals which are carried into the same point in R^ by the 303

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mapping ( 1 ) as constituting one coset. In other words, two functionals F l and belong to the same coset if and only if ( F i , 9 . ) = (F2,9.,),

1 <Ä
Obviously, the cyhnder set Ζ is the union of those cosets corresponding to the points of the set A, Conversely, any union of cosets is a cyhnder set in Φ'. T h e decomposition of Φ' into cosets is uniquely defined by specifying the Hnear subspace in Φ', consisting of those functionals which are carried into zero by the mapping ( 1 ) . Indeed, the condition

is equivalent to the condition (Λ-^2,φ,) =

X^k^n.

0,

Therefore two functionals belong to the same coset if and only if their difl^erence belongs to the subspace Ψ^. Note that the equations ( F , φ^) = 0 , 1 < Ä < w, imply that ( F , φ) = Q for any element of the form Φ = ^ΐΨΐ

+- +

^ηψη-

Therefore the subspace in Φ' can be defined as the subspace of those linear functionals F for which ( F , φ) = 0 for any element φ ΕΨ, where Ψ is the subspace in Φ generated by the elements φ^, φ^. T h u s we arrive at the following definition of a cyhnder set in Φ'. Let Ψ be a finite-dimensional subspace in Φ, and let denote the Unear subspace in Φ' consisting of those elements F for which (F,i/r) = 0

for

φεΨ.

T h i s subspace C Φ' is called the annihilator of the subspace Ψ. We decompose Φ' into cosets, putting into the same cosets all functionals which take the same values on Ψ (or, what is the same, all functionals whose differences Ue in the subspace Ψ^). We thus obtain the factor space Φ'ΙΨ^, whose elements are cosets. Associating with every functional F ΕΦ' the coset which contains it, we obtain a linear mapping of Φ' into Φ'/ΨΡ. Now choose any subset A C Φ'/ψ^. T h e collection of all elements

1.2

Basic Definitions

305

F ΕΦ' which are carried into elements of A by the mapping Φ' - > Φ'ΙΨ^ is called the cylinder set Ζ with base A and generating subspace Ψ^} This definition is more convenient to use than that given above, because it does not require that a basis φ^, 9n be given in the subspace Ψ.

1.2. Simplest Properties of Cylinder Sets Before studying the properties of cyhnder sets, we stop to consider some simple assertions concerning linear topological spaces. We will consider only locally convex linear topological spacejs, i.e., spaces in which every neighborhood of zero contains an absolutely convex neigh­ borhood of zero. The class of locally convex spaces is adequately broad; in particular, it contains all countably normed spaces. The following theorem on the extension of linear functionals holds for these spaces. Any linear functional F which is defined on a subspace Ψ of a locally convex linear topological space Φ can be extended to a linear functional on all of Φ. Indeed, the continuity of F implies that there exists a neighborhood U of zero in Φ such that | (F, φ) | < 1 iox ψΕίΙ ηΨ, Choosing an absolutely convex neighborhood of zero V C we take V as the unit sphere in Φ of a seminorm || φ || (i.e., we set || φ || = 1/sup | λ |, where λψ e F, for all ψ e Φ). Clearly j (F, φ) | < || φ || for all φεψ. By the Hahn-Banach theorem^ the functional F has an extension P, defined on all of Φ, which is additive and homogeneous and satisfies |(^, φ) \ < II φ II for all 9 G Φ. It follows that \{Ρ,ψ)\ < 1 for φ 6 F, which means that F is continuous relative to the topology of Φ. Next we show that if Φ is a locally convex linear topological space and Ψ is a subspace of Φ, then the space Φ'jW^ is the adjoint space ofW. Indeed, any element F ΕΦ' is a hnear functional on Φ, and conse­ quently on Ψ. Now two functionals F^ and F^ coincide on Ψ if and only if they belong to the same coset relative to i.e., if they correspond to the same element in the factor space Φ'/Ψ^. Thus, to every element ^ T h e notion of a cylinder set can b e i n t r o d u c e d for any linear topological s p a c e

Φ.

N a m e l y , let Ψ b e s o m e c l o s e d l i n e a r s u b s p a c e in Φ , a n d A s o m e s e t in t h e f a c t o r s p a c e Φ/Ψ.

T h e n it is n a t u r a l t o c a l l t h e c o l l e c t i o n o f e l e m e n t s φ 6 Φ s u c h t h a t t h e c o s e t w h i c h

contains φ b e l o n g s to ^

a cylinder set. H o w e v e r ,

Φ' c o r r e s p o n d i n g t o a n n i h i l a t o r s o f

all t h a t w e n e e d a r e c y l i n d e r s e t s

finite-dimensional

in

subspaces.

t T h e usual proofs of the H a h n - B a n a c h t h e o r e m are entirely valid for a s p a c e which has a s e m i n o r m , with respect to which the s p a c e need not b e

complete.

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Ρ Ε Φ'/ψ^ there corresponds a linear functional on Ψ, and to distinct elements of Φ'/ψ^ there correspond distinct functionals on Ψ. N o w let us show that every hnear functional on Ψ can be obtained in this way. L e t FQ be a linear functional on Ψ, T h e n , as we saw above, FQ can be extended to a hnear functional on all of Φ. T h e various possible extensions of FQ, since they all coincide on Ψ, belong to the same coset relative to Ψ^. T h u s , every linear functional on Ψ corresponds to some element of Φ'/Ψ^, which completes the proof. It follows from this result that if a subspace Ψ C φ is n-dimensional, then the factor space Φ'/Ψ^ is also n-dimensionaL Now let us consider cyhnder sets. A given cyhnder set may be defined by various generating subsets and bases. F o r example, if Φ

= ^ΐΨΐ

+

... + ^ηψη.

then the inequalities ( F , φ) ^ a and

define the same half-space in Φ'. Let us now clarify the conditions under which a cylinder set Z^, having generating subspace Ψ, and base A,, coincides with the cyhnder set Zg having generating subspace ΨΙ and base Fi^'st we note that the cylinder sets and Zg can be given by the same generating s u b s p a c e ¥^3. T h i s subspace is the annihilator of the subspace in Φ generated by the subspaces Ψ, and ¥^2» coincides with the intersection Ψ, η Ψ^. Since C ψ^, any coset with respect to ΨΙ belongs to some coset with respect to Ψ?. Associating with every coset with respect to ¥^3 that coset with respect to Ψ, which contains it, we obtain a linear mapping of the factor space Φ'ΙΨ^ onto the factor space Φ'/Ψ^. If we denote the inverse image of the set A, under the mapping by Ti\Ai)y then it is obvious that the cylinder set Z, can be defined by the generating subspace ΨΙ and the base Τΐ\Αι). It follows in the same way that the cyhnder set Zg can be defined by tlie generating subspace ¥^3 and the base 72X^2) (^2 denotes the hnear mapping from Φ'/ΨΙ onto Φ'/ΨΙ^ by which every coset with respect to Ψΐ is carried into that coset with respect to which contains it). Since evidently two cylinder sets with the same generating s u b s p a c e coincide if and only if their bases coincide, we obtain the following result: S u p p o s e that the cylinder sets Z j , Zg are defined respectively by the generating subsets and and the bases A, and A^. Set ΨΙ =

1.3

Basic Definitions

Π ¥^2· In order that that

307

and Zg coincide, it is necessary and sufficient T-\A^

= T-\A,\

(2)

where denotes the natural linear mapping of Φ'/Ψΐ onto Φ'/Ψ^, and Γ2 denotes the natural Hnear mapping of Φ'/ΨΙ onto Φ'/ΨΙ.^ We note also the following properties of cyHnder sets. (1) The complement of any cyHnder set is a cyHnder set. Indeed, if the cylinder set Ζ is defined by the generating subspace and the base Ay then Φ' — Ζ has the same generating subspace, and its base is the complement of A in the factor space Φ'/Ψ^. (2) The intersection of any two cylinder sets is a cylinder set. Indeed, we have seen that Z^ and Z2 can be defined by the same generating subspace in Φ'. Suppose that their bases are accordingly A^ and A2. Then Z i Π Z^ is the cylinder set with generating subspace and base Ai η A^.

The following property is proven in entirely the same way. (3) The sum of any two cylinder sets is a cylinder set. We see, thus, that the cylinder sets form an algebra of sets.^

1.3.

Cylinder Set Measures

We wiU henceforth consider only cylinder sets Ζ whose bases are Borel sets in Φ'ΙΨ^ (recaU that we are considering only generating subspaces such that Φ'ΙΨ^ is finite dimensional). If Z^, Zg, ... are cylinder sets with Borel bases, having the same generating subspace W^y then their union U^^i Z^ and intersection fl^^j Z^ are also cylinder sets with Borel bases. By a cylinder set measure in the space Φ' we will mean a numerical valued function μ(Ζ), defined on the family of all cyHnder sets with Borel bases, which has the following properties: (1)

0^μ{Ζ)

(2)

μ{Φ') = 1,

(3)

if the set Ζ is the union of a sequence Z^, Zg, ... of nonintersecting

2 Obviously,

< 1 for an Z ,

i f t h e f a c t o r s p a c e s ΦΊΨ\ a n d

Φ'ΙΨΙ a r e finite d i m e n s i o n a l , t h e n

Φ'ΙΨΙ

will b e a l s o . ^ A s y s t e m o f sets is c a l l e d a n a l g e b r a if it c o n t a i n s , a l o n g w i t h a n y t w o s e t s , their u n i o n and their

complements.

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cylinder sets having Borel bases and a common generating subspace Ψ\ then μ{Ζ) = I; μ(Ζ„), n=l

(4)

for any cylinder set Ζ (with Borel base)+

where U runs through all open cyhnder sets containing Z. A cyhnder set measure μ{Ζ) defines a measure on the Borel sets in every factor space Φ'ΙΨ^. Namely, if A is some Borel set in Φ'ΙΨ^, and Ζ is a cylinder set with base A and generating subspace Ψ^, then we set v^{A) = μ{ζγ

(3)

Obviously νψ is a positive normalized measure in Φ'which is regular in the sense of Caratheodory.* The measures induced by μ in diflFerent factor spaces Φ'\Ψ^ are not independent. If a given cyhnder set Ζ can be defined by the generating subspace ΨΙ and base A^ as well as the generating subspace Ψ\ and base then it is necessary that

because both sides coincide with μ{Ζ). Taking into account the condition indicated in Section 1.2 for two cyhnder sets, defined by diflFerent generating subspaces and bases, to t T h e following remarks m a y be useful. T h e authors are considering the weak called " w e a k dual*' o r " w e a k * " ) t o p o l o g y on

Φ\

Since any annihilator W

is a

(often closed

s e t i n Φ' in t h i s t o p o l o g y , e l e m e n t a r y f a c t s c o n c e r n i n g t h e d e f i n i t i o n o f a t o p o l o g y i n a factor space

Φ'ΙΨ^,

p l u s t h e fact t h a t t h e r e is o n l y o n e t o p o l o g y o n a

v e c t o r s p a c e (in o u r c a s e ,

Φ'ΙΨ^)

finite-dimensional

w h i c h m a k e s it a l i n e a r t o p o l o g i c a l s p a c e a n d s e p a r a t e s

p o i n t s , i m p l y t h a t a c y l i n d e r s e t is o p e n i n t h e w e a k t o p o l o g y o f Φ' if a n d o n l y if i t s b a s e is a n o p e n s e t i n

Φ'ΙΨ^.

W e observe also that condition

p o i n t e d o u t i n S e c t i o n 2.1 b e l o w , μ i s

finitely

4 i s s u p e r f l u o u s . I n d e e d , a s is

a d d i t i v e ; h e n c e Ζ C U i m p l i e s μ{Ζ)

B u t it is a s t a n d a r d r e s u l t o f m e a s u r e t h e o r y

that any

finite

(or even

Borel)

<

μ(υ).

measure

o n t h e B o r e l s e t s in / ? „ is r e g u l a r . A p p l y i n g t h i s t o t h e m e a s u r e s νψ ( s e e t e x t f u r t h e r a h e a d ) , w e s e e t h a t c o n d i t i o n 4 is s a t i s f i e d w h e n

U r u n s o v e r all c y l i n d e r s e t s w h i c h h a v e

the

s a m e generating s u b s p a c e as Ζ and whose bases are open sets containing the base of Z . * A m e a s u r e ν is c a l l e d r e g u l a r in t h e s e n s e o f C a r a t h e o d o r y , if f o r a n y B o r e l s e t one

has p{A)

where

=

infKC/),

U r u n s t h r o u g h all o p e n s e t s c o n t a i n i n g

A.

A

1.4

Basic Definitions

309

coincide, we can formulate the preceding equality in the following way. If Ψι C ¥^2» then for any Borel set A in the factor space Φ'/Ψ^ one has = ν^,[Τ-^{Α)1

(4)

where T'^A) denotes the inverse image of A with respect to the natural mapping Τ of Φ'/Ψ^ onto Φ'/Ψ^ {Τ carries every coset with respect to ¥^2 into that coset with respect to Ψχ which contains it). Thus, we have found a necessary condition for a system of measures νψ in the factor spaces Φ'/ψ^ to be induced by a cylinder set measure. This condition is also sufliicient. In other words, the following assertion holds. Suppose that {νψ{Α)} is a system of normalized positive measures, regular in the sense of Caratheodory, in the factor spaces Φ'/Ψ^, If Eq. (4) holds for every Borel set A in Φ'ΙΨΐ whenever C ψ^, then the measures ν ψ are induced by a cylinder set measure μ{Ζ) in Φ'. Indeed, for any cylinder set Ζ with generating subspace and base A we set μ{Ζ)

=

v^{A).

From (4) it follows that μ{Ζ) does not depend upon the manner in which Ζ is defined. Obviously μ{Ζ) is a cylinder set measure in Φ', and all the measures νψ are induced by μ. From now on we will call (4) the compatibility condition for the measures Ρψ. It can be shown that it is sufliicient to verify (4) only for half-spaces in Φ'. This assertion easily results from the following lemma: If the values of two positive normalized measures and Vg in a finite-dimensional space R coincide for all half-spaces in /?, then and identical. For the proof of this lemma cf. reference (29).

1.4. The Continuity Condition for Cylinder Set Measures We will henceforth consider only measures which satisfy a certain continuity condition. This condition is formulated in the following way: •

A cylinder set measure μ is said to be continuous, if for any bounded continuous function f{xi, x,„) of m variables, the function /(9Ί. ···. 9nd

= Jί

φ'

/{{Ρ, -Pi),

(F, Ψη,))

άμ{¥)

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is sequentially continuous^ in the variables φ^, G Φ. In other words, if hm^_,oo φα = 9jyj = 1, ^ , where the convergence is in Φ, then lim/(93,i, φi^) = / ( φ ι , φ^). We point out that the integral is well defined, because if Ψ denotes the linear subspace generated by 0 and any sequence { φ ^ such that lim^_,Qo = 0, we have lim^^^o μ{\ {F, ψχ) \ ^ A] = 0. In fact, let f{x) be a continuous bounded nonnegative function such that /(O) = 0 and f{x) = 1 for | Λ: | ^ A. By the continuity of μ we have l i m J / ( ( F , φ,)) άμ{Γ) = | / ( ( F , 0)) αμ{Ρ) = ο · ^ ( F ) = Ο But 7((F,φ,))^i^(F)>μ{|(F,φ,)| from which the assertion follows. The

converse is also true: If, for any ^ > 0 and any sequence ( φ ^

such that lim^._^oo Ψι = 0, one has lim/x{|(F,φ,)| ^ ^ } =

0,

then μ is continuous. For the proof, the reader is referred to a paper by Minlos [reference (49)]. T h i s enables us to use the following suflScient condition for the continuity of μ: Let μ^ be the measure on the real fine defined by /x^(—oo, Ö ) = /x{(F, φ ) < a}. If, for any sequence { φ ^ for which lim^_oo ψi = 0, and any bounded continuous function /(Λ:), one has limJ/(^)J^J^)=/(0), then μ is continuous. t In countably n o r m e d spaces, which have a metric topology, sequential a n d ordinary continuity are of course equivalent. T h e only place where sequential continuity intervenes is i n S e c t i o n 4 . 2 , w h e r e t h e u s e o f s e q u e n c e s a p p e a r s t o b e u n a v o i d a b l e .

1.5

Basic Definitions

311

Indeed, by the resuh mentioned above it suffices to shov^ that lim^{|(F, ^ } = 0 t-»oo

for any ^ > 0. But if f{x) is the function defined earher, then lim /x{|(F, ψ,)\ ^A}

A

= lim ^ , , ( [ - ^ , A]) < lim j f{x) άμ^^{χ) = 0.

1.5. Induced Cylinder Set Measures

L e t Τ be a continuous linear mapping of the hnear topological space 0 2 into the linear topological space Φ^. Denote by T" the mapping of Φί into 0 2 which is adjoint to Γ , i.e., the mapping such that (T"F, φ) = {Fy Τψ) for any φ e 0 2 and F e Φ[, Obviously, if Τ carries the finitedimensional subspace Ψ2 C 0 2 into the finite-dimensional subspace Ψι C 0 j , then T" carries the subspace into the subspace Ψ^. Indeed, suppose that FeWl T h e n ΤφβΨ^ for any ΦΕΨ^, and therefore {Fy Τφ) = 0. But this means that {TTyφ) = 0 for all φβΨ^, i.e., that

ΤΤΕΨΙ

T h u s we have proven that ΤΨζ C F r o m this it follows that the mapping 7" induces a mapping T[ of the factor space Φ^/Ψχ into the factor space 0 2 / ϊ ^ 2 > by which the coset F -\is carried into the coset ΤΨ+Ψ^ (in view of the inclusion ΓΨ^€ΨΙ the correspondence F + - > TT + Ψ2 does not depend upon the choice of representative F in the coset F + Ψ^), T h u s we have proven: If Γ is a continuous hnear mapping of 0 2 into 0 1 , then for every finite-dimensional subspace Ψι C 0 ^ there exsist a hnear mapping T[ of the factor space Φ[ΙΨΙ into the factor space Φ^Ι^Ι^ where ΨΙ denotes the annihilator of the subspace Ψι = ΤΨ^ in Φι, T h e mapping ΤΊ takes the coset F + ΨΙ into the coset TF + ^^2· S u p p o s e now that a cylinder set measure μι is given in 0 ^ . W e intro­ duce a measure on the cyhnder sets in 0 2 in the following way. S u p p o s e that the cylinder set in 0 2 is defined by the s u b s p a c e Ψ^ (in 0 3 ) a n d the base A^, L e t Ψι = ΤΨ^ and Αι = {Τί)-\Α2) C Φ^/ψ^ W e set μ2{Ζ2) =

μι{Ζι),

where is the cylinder set in Φ[ with generating subspace Ψι and base Ai. It is easy to see that μ2 is a cylinder set measure in 0 2 , and that in view of the continuity of Γ , ^ 2 satisfies the continuity condition. W e will call μ2 the measure induced from μι by the mapping T.

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For example, if Φ„^ is the completion of the countably Hilbert space Φ in the norm || φ then to every measure in Φ ^ there corresponds a measure μ in Φ' (and a measure /x^ in any space Φ^, where η > τη). We will call a measure in Φ' m-continuous if it is induced by a continuous measure in Φ^.

2 . T h e C o u n t a b l e A d d i t i v i t y of C y l i n d e r Set M e a s u r e s in

Spaces A d j o i n t t o N u c l e a r Spaces 2 . 1 . The Addltivlty of Cylinder Set Measures

Cylinder set measures have the following property of finite additivity: If Z i , Z ^ is a finite system of disjoint cylinder sets in Φ', then 1^{\J

Ζ,)=-^μ{Ζ,).

In fact, since, as was shown in Section 1 . 1 , we can find a common generating subspace for any finite system of cylinder sets, this assertion follows from the additivity of the measure νψ in the factor space Φ'/Ψ^. However, the measure μ does not by any means always have the property of countable additivity: it does not follow, generally speaking, that if a cyhnder set Ζ is the union of a countable family Z j , Zg, ... of nonintersecting cylinder sets, then μ{Ζ) =

%μ{Ζ,) k=l

(of course, the equality does hold if all of the Z ^ are defined by the same generating subspace). For us, however, it is essential that μ be countably additive. T h i s is connected with the fact that the class of cylinder sets in Φ' is rather narrow.^ Therefore it is natural to want to extend /x to a wider class of sets. T h i s class is the σ-algebra generated by the (Borel) cyhnder sets. As usual, by the σ-algebra of sets generated by the cylinder sets we mean the smallest class of sets which contains the cylinder sets and is closed under the operations of countable union and complementation. We will call the members of this σ-algebra the Borel sets in Φ'. ^ F o r e x a m p l e , t h e p o l a r o f a s e t ^ C Φ , g e n e r a l l y s p e a k i n g , is n o t a c y l i n d e r s e t i n Φ ' ( t h e polar

o f a s e t A i s t h e s e t o f a l l f u n c t i o n a l s F s u c h t h a t \ (F, φ) \ <

1 f o r all φ e

A).

2.1

Countable Additivity of Measures

313

T h e class of Borel sets is adequately broad; for example, if Φ contains a countable everywhere dense set of elements, then the polar of every set ^ C Φ is a Borel set in Φ'. In the case where the measure /χ, defined on the cylinder sets, is completely additive, it can be extended to all the Borel sets. T h i s extension can be carried out in the following way. We call the cylinder sets Borel sets of the zeroth class. S u p p o s e that Borel sets of class β have already been defined, where β is any transfinite number less than a. We call **Borel sets of class a* all countable unions of nonintersecting sets of class less than α and all complements of such unions. T h u s , Borel sets are defined for all transfinite numbers of the first and second classes. If

is a decomposition of a Borel set of class α into nonintersecting Borel sets of lower classes, then we set μ{Β) =

μ{Β,) k=l

and μ(φ' - β ) = 1 - μ ( β ) . Using the completely additivity of /x for cyhnder sets, we now show that starting from two decompositions

and Β = ^Βί

(or

Β = Φ' -

μ^Βή

of the Borel set Β into nonintersecting Borel sets of lower classes, we always obtain the same value for μ{Β). T h i s is easy to prove for sets of the first class: If

are two decompositions of a set Β of the first class into nonintersecting cyhnder sets, then :)·

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Indeed,

X KZu) = XX

μ{ζ, η ζ;) = XX

μ{ζ, η ζ;) = ν μ(ζ;.).

If now

Ä=l

Ä;=l

then

( Μ ^ · ) " ( Μ ^ > ) = *is a decomposition of Φ' into nonintersecting cylinder sets, and therefore

Χμ{Ζ,)

+

Χ μ { Ζ ί ) = 1 ,

i.e..

T h i s proves that /x is unambiguously defined on Borel sets of class 1. It can be shown that μ remains countably additive following this exten­ sion. For sets of higher classes the proof is carried out by means of transfinite induction. We remark that the extension of μ to the Borel sets in Φ' has the following property (regularity in the sense of Caratheodory): For any Borel set Β C Φ', μ{Β) = inf/x(Z), where Ζ runs through all countable unions of open cylinder sets Z^ such that Β C U^^i Zj,. T h e proof of this assertion is easily carried out by means of transfinite induction. We will see further on that there exist spaces for which every positive normalized cylinder set measure which has the continuity property is countably additive, and can therefore be extended to all the Borel sets. At the same time, there exist spaces in which not every measure can be extended to the Borel sets, but only measures satisfying certain additional conditions.

2.1

Countable Additivity of M e a s u r e s

315

The class of spaces for which any positive normalized cylinder set measure satisfying the continuity condition can be extended to the Borel sets is the class of spaces which are adjoint to nuclear spaces. T h i s result will be proven in Section 2.4. F o r the proof of this basic result we need certain results of measure theory. First of all we indicate the following simple criterion for the countable additivity of a measure. T h e o r e m 1 . In order that a measure μ on the cylinder sets in Φ ' be countably additive, it is necessary and sufficient that XM(Z*) = 1 for any decomposition Φ' = sets.

U^^i

of Φ' into nonintersecting cyhnder

Proof. T h e necessity of the condition follows directly from the definition of countable additivity. A s for the sufficiency, s u p p o s e that Ζ = \J]c=.i Zj^ is a decomposition of some cylinder set Ζ into nonintersecting cyhnder sets Z^, Zg, .... T h e n the space Φ' can be decomposed into the nonintersecting cylinder sets Φ' — Ζ , Z j , Zg, and therefore by the hypothesis of the theorem

, χ ( Φ ' - Ζ ) + Χ μ ( Ζ , ) = 1.

(1)

F r o m the finite additivity of μ it follows that μ{Φ' -Ζ)

+ μ{Ζ)=\.

(2)

Comparing (1) and (2), we obtain

k=l which proves the countable additivity of μ. T h i s theorem can be stated in another, equivalent, way. T h e o r e m 1'. In order that a measure μ on cylinder sets b e countably additive, it is necessary and sufficient that

litnKZ;) = 0

(3)

for any decreasing sequence Z^ 3 Z 2 3 ... of cyhnder sets whose inter­ section is empty.

316

Proof.

MEASURES

IN

LINEAR

TOPOLOGICAL

SPACES

Ch.

IV

Only the sufficiency of the condition needs to be proven. L e t

be a decomposition of Φ' into nonintersecting cyhnder sets. T h e n the cylinder sets

form a decreasing sequence with empty intersection, and so by hypothesis

limKZ;) = 0 . In view of the finite additivity of /x, this means that lim

1 -

2;

μ{Ζ,) =

0

k=l or, that Σ ^ = ι / χ ( Ζ ^ = T h e o r e m 1.

1. Consequently,

μ is countably additive

by

T h e o r e m Γ ' . In order that the measure μ be countably additive, it is necessary and sufficient that for any sequence {Zj.} of (not necessarily disjoint) cyhnder sets whose union is Φ', Χμ{Ζ,)^\, k=l

(4)

T o prove the sufficiency of this condition, we note that if the sets Zj^ whose union is Φ' are nonintersecting, then in view of the finite additivity of μ one has Χμ{Ζ,)^1.

(5)

On the other hand, inequality (4) is satisfied. Inequalities (4) and (5) imply k=l and therefore μ is countably additive by T h e o r e m 1. T h e necessity of the condition is obvious. Finally, we note that it is sufficient to require only that inequality

2.2

Countable Additivity of Measures

317

(4) hold for all sequences of open cylinder sets whose union is Φ'. T h i s follows at once from the fact that in view of the regularity of μ, for any cyhnder set Ζ we can find some open cylinder set whose measure exceeds that of Ζ by as little as desired.

2.2. A Condition for the Countable Additivity of Cylinder Set Measures in Spaces Adjoint to Countably Hilbert Spaces

T h e conditions for countable additivity given in the preceding section are inconvenient to apply. Here we introduce a condition for the count­ able additivity of measures on the cylinder sets in spaces adjoint to countably Hilbert spaces, which is more convenient to use. S u p p o s e that the cyhnder set measure μ in the space Φ' adjoint to a countably Hilbert space Φ is countably additive. T h e n , as we have seen above, it can be extended to all the Borel sets in Φ'. In particular, μ can be extended to all balls ^^^(JR) defined by inequalities of the form | | ί Ί Ι _ η < i ? . Indeed, SJ^R) consists of ah continuous hnear functionals on Φ such that φ)| < / ? if || φ ||^ < 1. Choose a countable set { φ ^ of elements which are everywhere dense in the unit ball — {|| φ L < 1} of the Hilbert space Φ^ and which lie in Π Φ. If we denote the strips \(¥,ψ^\ ^ R in Φ' by Aj,, it is obvious that Sn{R)=nA,,

i.e., .S^(/?) is a Borel set in Φ' (moreover, it is a Borel set of class 1). Therefore μ can be extended to every ball. Now let us show that for any € > 0 there is a baU 5^(/?), defined by an inequality of the form || F ||_^ < / ? , such that the μ measure of the com­ plement of Sn,{R) is less than € (assuming /χ. to be countably additive). Indeed, every element F ΕΦ' belongs to one of the spaces Φ^ and there­ fore satisfies some inequality of the form < R. Therefore Φ' is a countable union of balls, 00

Φ'=

ύΟ

U^U5„(Ä).

Since m ^ η implies that ^ ||/^||_n for any element ΡΕΦ', then 8γ^^ι) C 5^+i(« + 1). Consequently, Φ' is the union of an increasing sequence of balls S^Jji), i.e., Φ' =

U Sn{n)y

n=l

318

MEASURES IN

LINEAR TOPOLOGICAL SPACES

Ch.

IV

where S,(\)CS,{2)C,... Since μ{Φ') =

1, we have

But this shows that for any € > 0 there is an η such that the complement of Sγ^{n) has measure less than c. We have therefore proven the following assertion. T h e o r e m 2 . If is a positive normahzed countably additive cyhnder set measure in the adjoint space Φ' of a countably Hilbert space Φ, then for any e > 0 there is a ball S^iR) such that the /x-measure of any cylinder set Ζ lying outside S^{R) is less than e. Now we prove that the converse also holds. T h e o r e m 2'· S u p p o s e that μ is a positive normalized cylinder set measure on the adjoint space Φ' of a countably Hilbert space Φ. If for any € > 0 there is a ball S^iR) in Φ' such that the measure of any cylinder set lying outside is less than e, then μ is countably additive. For the proof of T h e o r e m 2' we need the following lemma. L e m m a 1 . F r o m any covering of a ball S{R) = {|| φ || < i ? } in a Hilbert space Η by open cylinder sets, one can extract a finite subcovering. A cylinder set in a Hilbert space Η is defined by the condition {{ψ>

...,(
where φχ, are fixed elements in / / , and A is some set in n-dimensional space Since (φ, φ^) is a hnear functional on i / , this definition agrees with that of Section 1. Since an open cyhnder set in a Hilbert space is an open set in the weak topology of Hy L e m m a 1 can be expressed briefly by saying that a ball in a Hilbert space is weakly compact^ (i.e., compact in the weak topol­ ogy)^ Astt

A l y i n g i n a t o p o l o g i c a l s p a c e X i s s a i d t o b e compact,

by open sets one can extract a t o p o l o g y i n t h e H i l b e r t s p a c e H,

finite

if f r o m a n y c o v e r i n g o f A

s u b c o v e r i n g . W e a r e c o n s i d e r i n g h e r e t h e "Weak

in w h i c h the n e i g h b o r h o o d s o f z e r o a r e d e f i n e d b y

inequalities of the form 9>fc)l < i . e . , t h e y a r e o p e n c y l i n d e r s e t s i n H.

1 <

Ä < «,

T h e c o m p a c t n e s s o f a s e t A in this t o p o l o g y m e a n s

that from any covering of A b y o p e n cylinder sets o n e can extract a

finite

subcovering.

2.2

Countable Additivity of M e a s u r e s

319

T h e proof of L e m m a 1 is as follows. With each element φ, || φ || < 1, we associate the interval — / ? < Λ: < i? of the real line, and denote these intervals by Ιφ, L e t / be the Tikhonov product of all of these intervals (see, for example, M . A. Naimark, N o r m e d Rings,*' Chapter 1, Section 2.12. Nordhoff, Groningen, 1959). Since the Tikhonov product of compact sets is compact, I is a compact set. N o w with each point e S{R)wt associate the point (φο> φ) on the interval Ιφ. We thus obtain a correspondence, to each e S{R)y of a point in the direct product of all the Ιφ, i.e., of a point in L A simple examination shows that the mapping {ψο, φ) is a homeomorphism of S{R) (with the topology induced in it by the weak topology in H) onto a closed subset cx(iS) of / . Since a closed subset of a compact space is compact, and α is a homeomorphism, then S{R) is itself compact. In other words, from any covering of S{R) by open cyhnder sets one can extract a finite s u b covering, which completes the proof. L e t us now prove T h e o r e m 2'. T o see that μ is countably additive, we have to prove, by the remark following T h e o r e m Γ ' , that if 00

is any covering of Φ' by open cyhnder sets, then

fc=l

Given € > 0, there exists by hypothesis a ball 5^(Ä) = {|| F \\_^ < R} in Φ' such that the measure of any cylinder set lying outside S^iR) is less than e. L e t Z^j, denote the intersection of the set Z^ with the Hilbert space Φ/^ in Φ'. It is obvious that the sets Z^^^ are open cylinder sets in Φγ^ which cover Φ^ and, consequently, cover the ball S^(/?). By L e m m a 1 we can choose a finite number of the Z^^, say Z^i, Z^^., which cover 5^(/?). Therefore we have Sn{R)C}^^Z^,C}J^Z,.

(6)

Let Ζ denote the cylinder set Φ' — U^^i Ζ^. It follows from (6) that Ζ hes outside S^iR) and its measure is therefore less than e. T h e n e > μ{Ζ) = μ ( φ ' -

J^J^Z,) > 1 "

K^,)

320

MEASURES IN

LINEAR

TOPOLOGICAL SPACES

Ch.

IV

(recall that the Z^. may intersect) and therefore

Since e is arbitrary, it follows from this that

X K Z . ) > 1, which shows that μ is countably additive, and completes the proof of T h e o r e m 2', T h e o r e m s 2 and 2' give a necessary and sufficient condition for a cylinder set measure in the adjoint space Φ' of a countably Hilbert space Φ to be countably additive. T h i s condition is that for any e > 0 it is possible to find a ball SJ^R) in Φ' such that the measure of any cyhnder set lying outside S^J^R) is less than e.

2.3. Cylinder Sets Measures in the Adjoint Spaces of Nuclear Countably Hilbert Spaces In this paragraph we shall prove a basic result concerning measures in the adjoint space of a nuclear countably Hilbert space. T h e statement of this result is as follows. T h e o r e m 3. S u p p o s e that Φ' is the adjoint space of a countably Hilbert nuclear space Φ. T h e n any positive normalized cylinder set measure μ in Φ', satisfying the continuity condition, is countably additive. We precede the proof of this theorem by certain lemmas on the connection between the measures of half-spaces and balls in w-dimensional space. Let /x be a positive normalized measure in an w-dimensional Euclidean space. We denote by /x(r, ω) the measure of the half space (jc, ω) > r, which is bounded by the plane perpendicular to the unit vector ω and situated a distance r from the origin {{x, ω) denotes the scalar product in the Euclidean space being considered). Further, we denote by μ{Ρ) the measure of a ball of radius R and center at the origin. In order to establish a connection between /x(r, ω) and μ{R)y we will consider not the measure /x(r, ω) but rather its average over ω. T h i s average is defined in the following way. A unit vector ω can be considered

2.3

Countable Additivity of Measures

321

as the radius vector of a point on the unit sphere. We introduce on the space Ω of these unit vectors the measure r, defined naturally as normalized (surface) measure on the unit sphere. In the spherical coordinates Xi = Ρ cos 9>i,

0 < ρ < 00, 0 < < TT, 1 < Ä < η - 2, Ο < ψη-ι < 2π,

= ρ sin
sin'^-''*

... sin φ„_2 άφ^ ... άφ^-χ.

(7)

If / ( ω ) is any function on the sphere Ω (or what is the same, a function of unit vectors ω), then by its average value we mean the integral / / ( ω ) ί / τ ( ω ) , which we wiU denote by < / ( ω ) > .

Thus

< / ( ω ) > - = ί /(ω)^τ(ω).

(8)

Let us now express the average of the measure of the halfspace {Xy ω) > r in terms of μ{Β). By definition we have =

/x(r, ω)

άτ{ω).

(9)

But μ(τ, ω) =

f(x, r, ω ) άμ{χ),

where /(Λ;, r, ω) denotes the characteristic function of the half-space (Xy ω) ^ r, i.e., the function which equals unity for (jc, ω) ^ r and zero for {Xy ω) < r. Substituting this expression for /x(r, ω) into ( 9 ) , we obtain =

f{x, r, ω ) ίί/ιι(Λ?)

άτ(ω).

F r o m this it follows that <μ(Γ, ω)> =

J φ{Xy r) άμ{χ\

where 9(Λ^» 0 =

ί /(^» ^. ω) ίίτ(ω).

322

MEASURES I N LINEAR

TOPOLOGICAL SPACES

Ch. I V

F r o m this formula one sees that φ{χ, r) equals the r-measure of the set of those vectors ω for which {x, ω) > r. Since the ends of these vectors form a segment on the unit sphere which is cut off from the sphere by the plane {x, ω) = r/| χ |, situated a distance r/| χ \ from the origin, then φ{χ.τ)=

f

^^^^^

{\~y^i-Uy,

if

r < \ x l

and φ(χ,τ) = 0

if

r ^ \ x \ .

F r o m this it follows that <μ(Γ, ω)> =

^,/^^"}

, - ί

άμ(χ) f

Τ\\η — φ) Υ π ^ τ<\χ\

(1 - fr-^

dy.

^ rl\k\

Since the expression (1 rl\x\

depends only upon Ψν

···>

| χ |, then changing to spherical coordinates p,

9 n - i > we obtain the formula ^00

(μ{τ,ω)}=

-

i ) V 7Γ

J

.1 J

r

{\-yη^--idydμ(pl

(10)

r/p

where, as stated above, μ{ρ) denotes the /x-measure of the ball of radius ρ and center at the origin. T h u s , we have estabhshed a connection between the average )> of the measures of the half-spaces (Λ:, ω ) > r and the measures μ{ρ) of balls. Since the function Φ{ρ)-=

Γ J r/p r/p

(l-y'Y^-^dy

is a monotone increasing function of p, we find from (10) that Π In) ^

Γ(^η -

Λ1

^ ) VTr J r/«

'^y[^ -

-^i^)!

for any r and Ä > r. It follows from this inequality that 1 -MÄ),

(11)

2.3

Countable Additivity of M e a s u r e s

323

where C(n, r/R) denotes the ratio

Now we show that if we set r/R = l / V n in C(n, r//?), then the set of numbers = C{n, Vn) will be bounded. T o show this, we observe that 2 J J (1 - yT-' On

dy

2 if-

[1 - ( / / n ) ] * - * dy

—

Therefore lim C„ =

2 f"exp(—i/)iiy — / - ^ ^ < J>xp(-i/)rfy

+00.

Since all of the C^y η = 1, 2, are positive and lim^_,Qo C^^ < + oo, the set { C J is bounded. L e t C = sup^ T h e n from (11) there follows the inequality

l - , ( i ^ ) < c ( , ( ^ Vn ,co)>. We have thus proven the following estimate μ{Η) by means of /x(r, ω ) .

lemma, which enables u s to

L e m m a 2 . L e t /LC be a positive normalized measure in an w-dimensional Euclidean space, let μ{R) be the measure of the ball of radius R and center at the origin, and let be defined by (9). T h e n 1-ΜΑ)<0·(μ(Α,α,)),

(12)

where C is a constant not depending upon either η or R. T h e half-spaces which are involved in L e m m a 2 are bounded by the planes (Λ:, ω ) = R/Vn which are tangent to the sphere with radius R/Vn and center at the origin. T o prove T h e o r e m 3, we need a lemma, similar to L e m m a 2, in which the sphere of radius R/Vn is replaced by an ellipsoid. We precede this lemma by the following assertion concern­ ing the average value of the square of the distance from the origin to the tangent planes of an ellipsoid.

324

MEASURES IN LINEAR TOPOLOGICAL SPACES

Ch.

IV

L e m m a 3. L e t r{w) be the (perpendicular) distance from the origin to the tangent plane of the ellipsoid

which is perpendicular to the unit vector ω . T h e n < r V ) > = η~^{λ\ + ... + λΐ).

Proof.

A simple calculation shows that τ%ω) = λ^^ω^ + ... + λ χ ,

where ω^, ..., are the coordinates of the vector.^ F r o m this it follows that

But since ωΐ -\- ... + equally distributed, then

=

^{
= I,

+ . . .+

xiwi).

and the coordinates

ω^,

are

and therefore

which proves the lemma. We now turn to the proof of the following lemma, which is (together with L e m m a 2 ) the central point in the proof of T h e o r e m 3 . ^ I n d e e d , i f t h e c o o r d i n a t e s o f t h e p o i n t o f t a n g e n c y a r e x[^\

xl^\ t h e n t h e e q u a t i o n

of the tangent plane to t h e ellipsoid has t h e f o r m

1

η

B r i n g i n g t h i s e q u a t i o n i n t o n o r m a l f o r m , w e find t h a t ω^. = τ{ω)χ'^^ΙλΙ a n d t h e r e f o r e

Λ:[0) = ωι,λΙΙν{ω). B u t (0)2

( 0 ) 2

+ ... +

a n d c o n s e q u e n t l y ^^(ω) = AJcoJ -f- . . . + λ^ω^.

= 1

2.3

Countable Additivity of Measures

325

L e m m a 4. L e t /x be a positive normalized measure in w-dimensional. Euclidean space i ? ^ , and let Q be an ellipsoid such that for each of its tangent planes, the measure of the half-space not containing the ellipsoid is less than €. T h e n the measure of the region outside any given sphere of radius R and center at origin which contains Q does not exceed C(€ + H^IR^)y where is the s u m of the squares of the semiaxes of Qy and C is a constant which depends neither upon η nor upon the choice of sphere or elhpsoid Q, Proof. Construct, for each unit vector ω, the plane (jc, ω) = R/Vη perpendicular to it which is tangent to the sphere of radius R/Vn. Parallel to each of these planes there is a plane (JC, ω) = r{w) which is tangent to the elhpsoid Q{r{w) is the distance from this plane to the origin). Certain of the vectors ω satisfy the inequality RjVn < r(a>); let Ωι be the set of these vectors. L e t ß g denote the sets of those vectors remaining, i.e., those for which R/Vn > r{w). We now show that the measure τ ( β ι ) does not exceed H^/R^ (recall that τ is the measure on the set of unit vectors which corresponds to the ordinary normalized measure on the surface of the unit sphere in / ? ^ ) . Indeed, it follows from L e m m a 3 that =

=

η But for ω G ß i we have ν\ω) which means that

'

f^(oj) dr ^

r

Γ2(ω)

dr.

^ Ω

^ R^tiy and therefore H^n

Hßi)

<

^

> τ ( β ι ) R^riy

(13)

·

Let, as usual, ^{R/Vw, ω) denote the measure of the half-space {Xy ω) > R/Vn. We wish to estimate its average value . Obviously

μ(A,,Δdτ+

(

μ{-^><Α

F r o m inequality (13) and the trivial estimate μ{R/Vny that

dr.

(14)

ω) < 1 it fohows

326

MEASURES IN LINEAR TOPOLOGICAL SPACES

Ch.

IV

On the other hand, for ω G ^he plane {x, ω) = R/Vn is further from the origin than is the tangent plane to the ellipsoid Q which is parallel to it. Therefore for ω e the half-space {x, ω) > R/\/n lies in the half-space {x, ω) > r{w). But by the hypothesis of the lemma the measure of the half-space defined by a tangent plane to the ellipsoid and not containing the ellipsoid does not exceed e. A fortiori we have μ{R|^/n,ω) < € for ωβΩ^, T a k i n g into account the trivial estimate τ{Ω^ < 1 , we obtain μ{RjVή,ω)dτ

< €.

(16)

2

F r g m ( 1 4 ) , ( 1 5 ) , and ( 1 6 ) follows the estimate <μ{Κΐνη,

ω)> < ^

+ e.

(17)

By L e m m a 3 , this implies that 1 - M Ä ) < C ( .

+ - ^ ) ,

(18)

where C is a constant not depending either upon n, or / ? , and 1 —μ{R) is the measure of the region outside the sphere of radius R and center at the origin. T h i s proves the lemma. We will show that a lemma similar to L e m m a 4 holds for spaces adjoint to a countably Hilbert space. We precede this lemma by the following remarks. L e t Φ' = U^.i b e the adjoint space of the countably Hilbert space Φ = CQ^i Φ^; let Ψ he a finite-dimensional subspace of Φ and its annihilator, and fix n. T h e natural mapping Τ of Φ' onto the factor space Φ'/Ψ^ induces a mapping of the subspace Φ'η into Φ'ΙΨ^. B u t every Φ^ is everywhere dense in Φ' (in the weak topology; cf. Chapter I, Section 3 . 1 ) and therefore the image of Φ^ in Φ'/ψ^ is everywhere dense in Φ'ΙΨ^, Since Φ'ΙΨ^ is finite-dimensional and the image of Φ^ is a linear subspace in Φ'ΙΨ^, it follows that Γ „ carries Φ^ onto Φ'/Ψ^. Therefore Φ'/Ψ'=Φ'η/Ψ''ηΦή·

Now denote by the orthogonal complement of Π Φ^ in the Hilbert space Φ^ (i.e., the collection of those ΡΕΦ^ such that ( F , F i ) _ ^ = 0 for all Γ^^ΕΨ^ Π Φ^). Obviously the natural mapping of Φ^ onto Φ'ΙΨ^ maps the subspace Ψ* one-one onto Φ'/Ψ^, Therefore any ΡΕΦ'

2.3

Countable Additivity of Measures

327

can be written in unique fashion as F = F ° + F * , where e Ψ^, F * G ' F * . We will call the orthogonal complement of in Φ;.+ We now proceed to the statement and proof of the analog of L e m m a 4. L e m m a 4'. L e t /x be a positive normalized cyhnder set measure on the adjoint space

φ'

=

Q

φ'

of a countably Hilbert space Φ = Π^^χ Φ^. L e t Q be an elhpsoid in the Hilbert space Φ^,* such that the s u m of the squares of its principal semiaxes is equal to H^, and the measure of any half-space in Φ', not containing Q, is less than €. If SJ^R) = { | | F | | _ ^ < R] is any ball in Φ ; containing then the measure of any cylinder set Z , lying outside Sn{R)y is less "than C(€ + H^jR^), where C is the same constant as in L e m m a 4. Proof. L e t Ζ be any cylinder set lying outside S^{R). L e t be its generating subspace, A C φ'ΙΨ^ its base, and denote by V'* the orthogonal complement of Π Φ^ in Φ^. If F is any element in Φ', then, as we have seen, it can be written in unique fashion as F = F » + F * , where F « G V^^^ F * G Ψ*. Denote by Ρ the m a p p i n g which takes F = F ^ + F * into F * , i.e., P ( F ) = F * . Since by construction is orthogonal to Π Φ^, then for elements F G Φ^, Ρ is the orthogonal projection of Φ^ onto ϊ ' * . T h i s orthogonal projection takes SJ^R) into a ball 5 * ( P ) in the s u b s p a c e and Q into an ellipsoid ^ * lying in Since the cyhnder set Ζ lies outside SJJi), its image Z * in hes outside 5 * ( P ) . We introduce a measure / i t * in Ψ*, defined by μ^{Χ)

= μ[Ρ-\Χ)]^μ{Χ

+

Ψ%

and apply L e m m a 4 to iS*(P), g * , and /x* in the finite-dimensional space Ψ*. For this we note that the s u m of the squares of the principal semiaxes of does not exceed H^, Indeed, the ellipsoid Q can be considered as the image of the unit sphere in Φ^ under a m a p p i n g Τ t F o r the case at h a n d , the foregoing s e e m s unnecessarily involved. η >

0 a n d F e Φ\

D e f i n e F*

e Φ'

t h e r e i s a u n i q u e φρ G Ψ s u c h t h a t ( F , ψ) =

b y ( F * , ψ) =

( φ , ^^jr)«; t h e n

^ F

-

F * 6

Indeed,

( φ , 0^)n f o r a l l ψ and F

=

F» +

given ΒΨ. F * is

the desired decomposition. * B y a n e l l i p s o i d i n t h e H i l b e r t s p a c e Φ η w r e m e a n t h e i m a g e o f t h e u n i t s p h e r e || F b y s o m e l i n e a r m a p p i n g T.

<

1

T h e condition that the series consisting of the s u m of the

s q u a r e s o f t h e p r i n c i p a l s e m i a x e s o f t h i s e l l i p s o i d c o n v e r g e m e a n s t h a t Γ is a H i l b e r t Schmidt operator.

328

MEASURES IN

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TOPOLOGICAL

SPACES

Ch.

IV

whose Hilbert-Schmidt norm equals H, T h e elhpsoid is the image of the unit sphere in under the mapping PT. But the H i l b e r t Schmidt norm of PT does not exceed H.^ Consequently, the s u m of the squares of the semiaxes of 5 * does not exceed H^. L e t us now show that the measure of any half-space in which does not intersect 5 * is less than €. Indeed, if C * is a half-space in which does not intersect ρ * , then C = C * + is a half-space in Φ' which does not intersect the elhpsoid Q, and therefore ^*(C*) =

= ^(C) < e.

+ η

Finally, since Q C S^{R\ then ρ * C Therefore, in view of L e m m a 4, the /x*-measure of the region outside does not exceed C(e 4 But the b a s e y l * o f Z l i e s + outside and therefore

Let us now proceed to our main goal—the proof of T h e o r e m 3 . In other words, we wish to prove that a positive normalized cylinder set measure in the adjoint space of a nuclear space is countably additive. Proof of Theorem 3 . As has been shown in T h e o r e m 2 ' , to prove the countable additivity of μ it suffices to show that for any € > 0 one can find η and R such that the measure of any cylinder set lying outside the ball SJ^R) = {|| F ||_^ < K) is less than e. First we use the continuity condition imposed upon μ. It follows from this condition that there exists a ball SJ^p) = {\\F\\_^ ^ p} such that the measure of any halfspace in Φ' which does not intersect S^{p) has measure less than € / 2 C , where C is the constant in L e m m a Since the space Φ is nuclear, there is an η such that the ball considered in the Hilbert space Φ^, is an ellipsoid, the s u m of whose ^ I n f a c t , i f Λ,Λ,

IPTII Since

Ρ

t ^ *

... i s a n o r t h o n o r m a l b a s i s i n Φ ^ , t h e n

^\\PTf,\\'

<\\P\\

i s a p r o j e c t i o n o p e r a t o r , || is t h e i m a g e o f A

= IIPIlll T\\, = HHP!!

^\\Tf

P|| =

1 a n d t h e r e f o r e ||

ΡΓΙΙ2 <

H.

under the one-one correspondence between

mentioned just preceding L e m m a

Ψ*

and

Φ'/ψ^

4\

t t I n d e e d , since sequential a n d o r d i n a r y c o n t i n u i t y a r e e q u i v a l e n t in a n u c l e a r s p a c e , f o r a n y δ, ^ that

> 0 t h e r e is a n e i g h b o r h o o d U o f z e r o in Φ ( s a y , U =

^ { | ( F , φ)\ >

{llPII-m <

A}

<

8 f o r φΕίΙ.

Choose

any ^

>

0

and

take

{|| φ II,» < δ =

ο)

such

e/2C;

then

is t h e d e s i r e d ball, w h i c h follows f r o m t h e o b s e r v a t i o n that a n y half-

s p a c e w h i c h d o e s n o t c o n t a i n t h e z e r o f u n c t i o n a l c a n b e w r i t t e n a s {(P, φ) > the existence of n o n z e r o functionals F for which

(P, ψ) = ||Ρ||_« II Φ IL-

A},

and from

2.3

Countable Additivity of Measures

329

principal semiaxes is finite (here η must be chosen so that the mapping of into Φη is H i l b e r t - S c h m i d t ) . L e t denote the s u m of the squares of the principal semiaxes of the ellipsoid S„^(p) in Φ^, and choose R so large that the ball S^iR) in Φ^ contains the elhpsoid 5^^(ρ), and also (19) By L e m m a 4', for any cyhnder set Ζ in Φ', lying outside S ^ ( i < ! ) , one has the estimate , ( Z ) < C ( ^ +

^ ) .

It follows from (19) that μ{Ζ) < €. T h u s , we have found a ball S^iR) such that the measure of any cylinder set Ζ which lies outside S^iR) has /x-measure not exceeding the given value € > 0. Hence T h e o r e m 2' imphes that the measure μ is countably additive. T h i s concludes the proof of T h e o r e m 3. It can be shown that the nuclearity of the space Φ is not only a suflii­ cient, but also a necessary condition for every cylinder set measure in the adjoint space Φ' to be countably additive. T h i s assertion is proven by constructing, for any nonnuclear countably Hilbert space Φ, a positive normahzed cylinder set measure in Φ' which is not countably additive. Moreover, this measure can be chosen to be a so-called G a u s s i a n measure (cf. T h e o r e m 2 of Section 3). We have already mentioned that in certain questions one can consider, instead of nuclear spaces, two Hilbert spaces which are connected by a nuclear mapping. L e t us indicate here the analog of T h e o r e m 3 which is obtained by this replacement. T h e o r e m 4. L e t and / / g be Hilbert spaces, and let Γ be a nuclear mapping of i / g into H^. S u p p o s e that /x is a positive normalized cylinder set measure, satisfying the continuity condition, in the adjoint space H[ of T h e n the measure in H2 induced^ by Τ and μ is countably additive. T h e proof of this theorem is a word-for-word repetition of the proof of T h e o r e m 3. T h e point is that in proving T h e o r e m 3 all that was used was that μ is continuous in one of the Hilbert spaces Φ ^ and that there exists a nuclear mapping of Φ ^ into one of the spaces Φ,^. We note that while T h e o r e m 3 talks about all measures in Φ', T h e o r e m ^ C o n c e r n i n g a m e a s u r e i n d u c e d b y a m a p p i n g , cf. S e c t i o n

1.4.

330

MEASURES

IN

LINEAR

TOPOLOGICAL

SPACES

Ch.

IV

4 concerns only measures in / / g which are induced by some measure in H[. 2.4. The Countable Additivity of Cylinder Set Measures In Spaces Adjoint to Union Spaces of Nuclear Spaces T h e theorem proven in the preceding section is not adequate for certain applications. F r o m example, this theorem does not apply to measures on the cyhnder sets in the conjugate space K' of the space Κ of all infinitely diflFerentiable functions having bounded supports. T h i s is due to the fact that Κ is not a countably Hilbert space, but rather the union space of its nuclear subspaces Κ{α), consisting of those functions in Κ wrhich vanish for | Λ: | < a. N o w every s u b s p a c e K{a) is closed in Ky and a sequence {^^(Λ:)} of functions in Κ converges to zero if and only if all of the ψγ,Χχ) belong to some one subspace K{a) and converge to zero in the topology of K{a). Moreover, from the general form of linear functionals on K{a) it follows that every linear functional on K{a) can be extended to all of K, In order to obtain a theorem on the complete additivity of measures on cyhnder sets which is also useful for the space Κ\ we introduce the concept of the union space of hnear topological spaces. L e t Φ^^^ C Φ<2) c ... be an increasing sequence of linear topological spaces such that the topology in each Φ^"^^ coincides with that induced in it by φ<"»+ΐ). We call the hnear space Φ = U^^i Φ^^^ the union space of the Φ^^\ T h e space Φ is not necessarily a linear topological space; however, we will say that a sequence {ψ^ of elements in Φ converges to zero, if all of the belong to some one Φ^^\ and {φ^} converges to zero in Φ^'^Κ An additive homogeneous functional on Φ will be called linear^ if its restriction to each Φ<^^ is a hnear functional on Φ^^\ Conversely, we will assume that any linear functional F^'^^ on Φ^'^^ can be extended (not necessarily uniquely) to a hnear functional F on Φ. T h u s , every hnear functional F^'^^ on Φ^"^^ can be considered as the restriction to Φ<^^ of some hnear functional F on Φ. T h e functionals F^^\ F^^\ ... corresponding to a given functional F on Φ are compatible in the follow­ ing sense: if m < w, then for any element ψ e Φ<^^ we have {F^'^\ ψ) = {F^'^\ φ), since both sides coincide with ( F , φ). T h u s we see that an element F of the conjugate space Φ' can be written in the form F = {Fti>,F<2), . . . } ,

where

e Φ^^^^', and the functionals F^^\ F^^\

... are compatible.

2.4

Countable Additivity of M e a s u r e s

We now consider cylinder sets in Φ'. It is easy to see that set is defined by a cyhnder set Z^"^^ in one of the spaces consists of all elements F = F^^\ ...} of the space Φ' pim) e ^ ( m ) ^ill denote the cyhnder set in Φ', defined by set Z<^> in

by

331 any such Φ^'^^' and for which a cyhnder

ZiZ^"^^).

It is easily verified that the weak neighborhoods in Φ' coincide with the cylinder sets in Φ' defined by the weak neighborhoods in the spaces Further on we will use the m a p p i n g s of Φ' into Φ^^^\ which carry an element F = {F^^\ F^^\ ...} of Φ' into its mth coordinate F^'^K F r o m the compatibility of the F^'^^ it follows that for m < w we have Tmf^TnyiFin) = fim)^ Therefore, setting = Γ ^ ( Γ ^ ) - ι , we obtain a mapping of the space Φ^^^' onto φ^^^\ by which every functional F^'^^ is carried into a compatible functional F^^K L e t us now consider a measure /χ on the cylinder sets in Φ'. T h i s measure defines a measure on the cylinder sets in every one of the Φ^^^, Indeed, we set μΛζη

=

μ[Ζ{ζη]

for any cyhnder set Z^^^ in Φ<^>', where Z{Z^^^) is the cylinder set in Φ' defined by Z^^K Obviously j i t ^ is a measure on the cylinder sets in Φ<'»>'. T h e measures μ^^ ... thus obtained are compatible in the sense that for m < w and any Z^^^ in Φ<^>' we have ,xJZ^->) = μ,[{Τΐγ'Ζ'-'],

(20)

Indeed, it is easy to see that the sets Z<"*> and ( Γ ^ ) - ^ Ζ < ^ > define the same cylinder set Ζ in Φ', and therefore both sides of (20) coincide with /x(Z). T h e converse is also true: if /χ^, /xg, is any sequence of compatible measures on the cylinder sets in the spaces Φ^^^', Φ^2)'^ ^ then there exists a measure μ on Φ' such that

/^[ΖίΖ*"·))] = ^.«(Ζ"»)) for all m and all cyhnder sets Z^^^ in Φ<'^>'. Of course, since the measures μ^ are finitely additive, μ will also be finitely additive. With somewhat greater effort, one can show that the countable additivity of each of the /x^ implies, under certain assumptions regarding /x, the countable additivity of μ. T h a t is, the following theorem holds. T h e o r e m 5. L e t Φ be the union space of the countably normed spaces Φ^^\ Φ<2), .... S u p p o s e further that /x is a positive normalized

332

MEASURES

IN LINEAR

TOPOLOGICAL

SPACES

Ch.

IV

measure on the conjugate space Φ\ such that for every m the measure induced by μ on Φ^^^' is countably additive and regular in the sense of Caratheodory. T h e n μ is also countably additive. Proof. According to T h e o r e m 1 w e have to show that any decreasing sequence Z j 3 Zg 3 ... of cyhnder sets Zj^ in Φ', such that lim;^^oo K^k) ^ € > 0, has nonempty intersection. N o w each of the Z,^ is defined by a cylinder set Z<^*) in Φ<^*)'. If the set of aU nij, is bounded, then without loss of generality we may suppose that τη^ = = ... = m. In this case the nonemptiness of Π ^ ^ ι Zy^ follows from the countable additivity Now assume that the are unbounded. In this case we may without loss of generahty suppose that < < .... By hypothesis each of the μ^^ is regular in the sense of Caratheodory. Therefore in each of the spaces Φ^^^^^' there is a weakly closed set such that ß^ C Z^^^^ and

Further, by T h e o r e m 2 the countable additivity of μ^^ imphes the existence in each φί'"*)^ of a weakly compact set (a ball) Q such that ^^mS^k) > 1 - e / 2 - < * ^ + 2 ) Obviously D„ = B„ Π is also weakly compact and

Set

It is easy to see that

C Z'""', and

Therefore μ^J^EJ^) ^ ^e. It is further obvious that each of the Ej^ is weakly compact in Φ<^*)', and TZ'fEj, C E^ f o r ; < k. We now choose an element from each of the £^ (the Ej, are non­ empty, as μ,nJ i e ) . F o r ; < nij, set k

rrij

Since T^liE^,) C E^, the elements F[^\ F^^\ ... belong to the weakly compact set E^. One can therefore choose a subsequence pa)

pa)

2.5

Countable Additivity of Measures

333

which converges weakly to some element F,^^ in Φ^^0\ N o w we consider the elements Fl^\ Fl^\ ... (we may clearly assume that > 2) which lie in (since T^* Ej^ C £"2). Since E^ is weakly compact, we can find a weakly convergent subsequence of [F^^^] which has limit F^^^ in Φ^^^^'. F r o m the weak continuity of TZ\ (which is easy to show) it follows that T^lF^^ = F^^. In similar fashion we construct elements F^^, F^^, ... in φ ( ^ ' ^ 3 ) ' , φ ( ^ 4 ) ' , ..., such that F^. = T^^^m^ for J < and we set Pm = TZ^'Fm^ for m < nij,. T h e n F = [Ε^,^^..,] is an element in Φ', and moreover belongs to each of the Zj^, since F^^^ ^ Ej^C Z^^^K T h i s proves that the sequence Ζχ, Zg, ... of cyhnder sets in Φ' has nonempty intersection, and therefore that μ is countably additive, which completes the proof. Since, according to T h e o r e m 3, any positive normahzed cyhnder set measure, in the conjugate space of a nuclear space, which satisfies the continuity condition is countably additive, we obtain from T h e o r e m 5 the following result. T h e o r e m 6. S u p p o s e that Φ is the union space of an increasing sequence Φ^^^ C φ<2) c ... of nuclear spaces. T h e n any positive normahzed measure, on the cyhnder sets of the conjugate space Φ', which satisfies the continuity condition is countably additive. Since the space Κ is the union space of its nuclear subspaces K{n), any measure on the cyhnder sets of K' which has the properties indicated above is countably additive.+ 2.5. A Condition for the Countable Additivity of Measures on the Cyhnder Sets in a Hilbert Space T h e o r e m 3 gives a condition for the countable additivity of all cylinder set measures in the conjugate space of a countably Hilbert space. In this paragraph we indicate a condition for the countable additivity of a specified measure μ defined on a Hilbert space H. Let B i , ... be a sequence of positive-definite operators in / / , with domain H, By means of these operators we introduce a new topology in i / , taking as a base of neighborhoods of zero the system of sets U^{R) in i / , where U^(R) consists of Β\1 φ e Η satisfying (β^φ, φ) < R^. We will call this topology the topology defined by the operators B^, B^, .... T h e following theorem holds. t I t is p e r h a p s

worth

the theory of generalized

mentioning explicitly that

this

r a n d o m processes as does the

in t h e t h e o r y o f o r d i n a r y r a n d o m

processes.

result

plays

Kolmogorov

the

same

extension

role

in

theorem

334

MEASURES IN LINEAR

TOPOLOGICAL

SPACES

Ch.

IV

T h e o r e m 7 . In order that a positive normahzed measure /x on the cyhnder sets' in the Hilbert space Η be countably additive, it is necessary and sufficient that μ be continuous relative to the topology in Η defined by some sequence B^, B^y ... of positive-definite nuclear operators. T h e continuity of μ means the following: F o r any e > 0 there exists a δ > 0 and η such that the inequality (β^φ, φ) < δ implies that /χ(Γ^) €, where denotes the strip defined by \{ψ,φ)\ ^ 1. Proof. First we prove the necessity of the condition. S u p p o s e that μ is countably additive. We construct for any w > 0 a positive-definite nuclear operator such that the inequality (ß^9?, ψ) < 1 βη imphes that μ{Trf) < I In. T h i s operator is constructed in the foHowing way. Since ^ is countable additive, it can be extended to all balls in H. B u t Η is the union of a countable family of increasing balls, and therefore there is a ball S{R) = {\\ φ || < /?} such that the measure of its complement is less than 1 /In. We define Bγ^ by setting (Βηφ^φ)=

(

\{ψ^φ)\^άμ{φ).

Obviously is positive-definite. T o show that it is nuclear, we note that for any orthonormal basis {φt^ in Η one has

X ( 5 n 9 . , 9 . ) =

ί

%\{ψ^.Φ)?άμ{φ) \\φ\\^άμ{φ)^Κ'

ί

SkR)

άμ{φ)^ΚΚ

^ SiR)

In other words, the series Σ^=.ι {Bγ^φf^y converges for any orthonormal basis {φf^}. A s was shown in Chapter I, Section 2.3, it follows that B^, is a nuclear operator. Now consider any element φ such that {Β^,φ, φ) < 1 /2n, and let u s estimate the measure of the strip defined by K ^ , 0)1 > 1. Obviously μ{τ,)

= μ(τ;) +

μ(τ-ι

where is that part of contained in the ball 5 ( / ? ) , and Γ ^ ' is that part lying outside S{R). In view of the choice of S{R) we have /χ(Γ^') < l/2n. ' B y a cylinder set in Η w e m e a n t h e collection o f those e l e m e n t s φ f o r w h i c h Ψl)>'^'Λφ,Ψk))G w h e r e 9>i,

A

φ* a r e e l e m e n t s o f H, a n d ^ i s a B o r e l s e t i n ^ - d i m e n s i o n a l E u c l i d e a n s p a c e .

S i n c e (
3.1

G a u s s i a n Measures

335

On the other hand, from the inequality |( which holds for all φ Ε Γ φ and therefore for all φ G Γ ^ , it follows that μ(Γ;)= f

ί

\{φ,φ)\^μ{φ)

\{ψ,φ)\^μ{φ)

=

{Βηψ,ψ)^^.

S(R)

Hence /χ(Γ^) < 1/η. T h u s we have constructed a sequence B^, B^, ... of positive-definite nuclear operators such that the inequality φ) < l/2n implies μ{Τ^ ^ l//z, where is the strip defined by \{ψ,φ)\ ^ 1. T h i s means that μ is continuous relative to the topology in Η defined by the Bj^, and proves the necessity of the condition of the theorem. T h e sufficiency of the condition can be proven by using L e m m a 4. We omit the details of this proof.

3. Gaussian Measures in L i n e a r T o p o l o g i c a l Spaces 3.1. Definition of Gaussian Measures We will consider here Gaussian measures in linear topological spaces. First we describe Gaussian measures in the finite-dimensional case. Let be an n-dimensional linear space, in which is defined a scalar product {x, y). T h i s scalar product defines a metric in and, in particu­ lar, defines L e b e s g u e measure in i?^. We introduce the G a u s s i a n measure in i?^ corresponding to the scalar product (Λ;, y), setting

'^(^) =

( 2 ^

In other words, Gaussian measures in an n-dimensional linear space are always defined by means of scalar products. If ^ is a non-degenerate linear transformation in and. (Λ:, y) is a scalar product in JR^^, then (x, y \ = {Ax, Ay) is also a scalar product. T o this scalar product there corresponds the G a u s s i a n measure

(27r)^^J X

where d^x denotes the L e b e s g u e measure corresponding to the scalar

336

MEASURES

IN

LINEAR TOPOLOGICAL SPACES

product ( j c , It is easy to see that d^x measure μ, can be written in the form μ,{χ)

=

det A

Ch. IV

| det ^ | dx. Therefore the

I

We note the following lemma concerning Gaussian measures in finite-dimensional spaces, which we wiH use further on. L e m m a 1 . Let R^^ be an w-dimensional Euclidean space with scalar product {x, y)y and R,,^ an m-dimensional subspace in R^^. L e t μ^^ be the Gaussian measure in R^ corresponding to the scalar product {Xy y), and denote by μ,,^ the Gaussian measure in R^^ corresponding to the same scalar product. T h e n for any subset X of R^^^ we have the following compatibility condition between μ „ and /χ„^: μ.η(^) = ^n[Q-Kx)l

(2)

where Q denotes the operator of orthogonal projection of

onto R„^.

Proof. L e t R ^ - m denote the inverse image of the origin with respect to Q. Obviously is the orthogonal s u m of the subspaces and Rn-my therefore any Λ: e i?^ can be written in unique fashion as X = x' x'\ where x' e R^,^ and x" e R n - m - It is also obvious that {x,y) = {x\y')

+

(x'\y"\

and that the L e b e s g u e measure dx in R^ is the product of the L e b e s g u e measures dx' in R^^^ and dx" in Rn_,y^y defined by the scalar product (A:, y). Since the set Q~\X) is the orthogonal s u m of X and we have

^

1

Since 1 (2^)i"

g-Hx',x')

dx',

to prove (2) it suffices to verify that 1 {Inf (n-m)

(3)

3.1

Gaussian Measures

337

But in coordinate form, the integral in (3) is ^

^expl-liW'f

+

... +

K-rnf]]

dx{' ... dx-_,,.

(3')

Since txp{-^x^)dx

=

νίπ,

the integral (3') is equal to 1, which proves (2). Along with the Gaussian measures just considered, so-called improper (or degenerate) Gaussian measures can also be considered. T h e s e are defined by a formula of the form

where i ? ^ is an m-dimensional hnear subspace of {Xy y) is a scalar product in 7 ? ^ , and dx is the L e b e s g u e measure in defined by this scalar product. Let us now proceed to the construction of Gaussian measures in the infinite-dimensional case. T h e s e measures will be constructed in the conjugate space Φ' of a locally convex linear topological space Φ. As we saw in Section 1.1, the local convexity of Φ guarantees that any linear functional on any subspace Ψ oi Φ can be extended to a hnear functional on all of Φ. In addition, the conjugate space Ψ' of any finite-dimensional subspace Ψ of Φ is isomorphic to the factor space Φ'/Ψ^, where denotes the annihilator of Ψ (i.e., the collection of all linear functionals on Φ such that {Fy φ) = 0 for φ e Ψ), Gaussian measures in an infinite-dimensional space Φ' are defined by means of scalar products Β{φ, φ) defined in Φ. T h u s , suppose that B{ψy φ) is a nondegenerate scalar product in a real locally convex linear topological space Φ, continuous in the topology of Φ. First we define a Gaussian measure in every finite-dimensional subspace Ψ of Φ by means of the scalar product β ( φ , φ)y and then we carry over these measures to the factor spaces Φ'/Ψ^. We define a measure τψ in each w-dimensional subspace Ψ oi Φ by

where dφ is the L e b e s g u e measure in Ψ corresponding to the scalar product Β{φ, φ). In view of L e m m a 1 these measures are compatible in the following sense: If C ψ^, then for any Y C we have =

r,[Q-i\y)l

(6)

338

MEASURES

IN LINEAR

TOPOLOGICAL

SPACES

Ch.

IV

where denotes the operator of orthogonal projection of Ψ2 onto Ψι (relative to the scalar product Β{φ, φ)), and t j , are the measures Τψ^ and Τψ^. We remark that the finite-dimensional Euclidean space Ψ (with scalar product β ( φ , φ)) is isomorphic to its conjugate space But, as was pointed out above, the space Ψ' is isomorphic to the factor space Φ'/Ψ^. We have thus established a natural isomorphism A ψ between Ψ and Φ'/Ψ^. I n view of this isomorphism there corresponds to the measure Τψ ιηΨ ζ measure νψ in Φ'/Ψ^, defined by νψ{Χ) = τψ[Α-\Χ)1

Χ€Φ'/Ψ^,

(7)

Let us show that the measures νψ define a measure on the cyhnder sets in Φ', i.e., that they are compatible. For this, according to Section 1 . 3 , it suffices to show that if C ψ^, then for any set X C Φ'/Ψζ one has^ v,{X) = v,[Q-\X)l

(8)

where Q denotes the natural mapping of Φ'/ΨΙ into Φ'/¥^ί. In view of ( 7 ) , we can write ( 8 ) , which is to be proven, in the following form: r^[A-\X)] = r,[A-^Q-\X)] (we have denoted Αψ^ and Αψ^ by Ai and A2 respectively). If we set Αΐ\Χ) = y , then this equation becomes τ , ( Υ ) = r,[A-^Q-^A,{Y)].

(9)

In view of relation (6) it remains to show that the m a p p i n g 5 i = A^^QA^ is the orthogonal projection of onto Ψ^. B u t this follows directly from the fact that Al^QA^ acts on elements φ^Ψ^ according to the scheme φ—^Α,φ

+ Ψ',—^Α,φ +

ΨΙ—^Φ. Λ-

and on elements φ in the orthogonal complement of

in

according

t o 3

φ ^

>Α.ψ + '

>

^ 2

Q

> ^

0.

Λ-^

T h i s proves that the measures νψ are compatible. ^ With every element

φ € Ψ we a s s o c i a t e t h e l i n e a r f u n c t i o n a l F ^ , , d e f i n e d b y Fy,{φ)

=

(B φ, φ). F r o m t h e n o n d e g e n e r a c y of Β {φ, φ) i t f o l l o w s t h a t t h e i m a g e o f Ψ i s a l l o f Ψ\ ^ W e d e n o t e h e r e νψ^ b y V J , νψ^ b y VG» a n d f u r t h e r τψ^ b y 3 S i n c e Β{ψ, φ) = 0 , V e Ψ^, f o r s u c h
a n d τψ^ b y T J .

a n d t h e r e f o r e Α^ψ - f

3.2

Gaussian Measures

339

We denote by μ the measure defined on the cyHnder sets of Φ' by the F r o m the fact that the scalar product φ) was a s s u m e d to be continuous in the topology of Φ, it follows easily that the measure μ satisfies the continuity condition. We omit the simple proof of this statement, which involves writing down the measures in coordinate form. One can remove the condition of nondegeneracy that was imposed upon Β(φ, φ). A s s u m e that there are nonzero elements ψ for which Β{φ, φ) = Ο, and let X be the totality of all such φ. T h e n is a linear subspace of Φ which is closed in view of the assumed continuity of φ). L e t be the subspace in Φ' consisting of all continuous linear functionals F which vanish on X, T h e n X^ is the conjugate space of the factor space Φ / Χ , and Β{φ, φ) induces a nondegenerate scalar product Βι{φ, φ) on the latter. We can therefore construct the Gaussian measure μι in X^ defined by the scalar product Βι(φ, φ). T h e measure μ in Φ', defined for any cyHnder set Y in Φ' by μ{Υ)=μ,(ΥηΧ^). is called a Gaussian measure in Φ'. S u c h G a u s s i a n measures, which are concentrated on subspaces of Φ', are called degenerate or improper. Let us now stop to consider the case where Φ is a Hilbert space, and the Gaussian measure μ is defined by the scalar product (φ, φ) in Φ. In this case the spaces Φ and Φ' can be identified, and we can s u p p o s e that μ is defined in the space Φ itself. T h e cyHnder sets Ζ in Φ are the or­ thogonal s u m s of subspaces of Φ, having finite-dimensional orthogonal complements A, with (Borel) sets X lying in A. T h e measure of such a cyHnder set is given by (2π) where (φ, φ) is the scalar product induced in the finite dimensional subspace A by the scalar product in Φ, and άφ is the corresponding L e b e s g u e measure in A.

3.2. A Condition for the Countable Additivity of Gaussian Measures in the Conjugate Spaces of Countably Hilbert Spaces In T h e o r e m 4 of Section 2, it was shown that if a measure μι in a Hilbert space Hi satisfies the continuity condition and if T" is a H i l b e r t -

340

MEASURES IN

LINEAR

TOPOLOGICAL

SPACES

Ch.

IV

Schmidt mapping of into a Hilbert space H^y then the measure /xg in i / 2 > induced by Τ and / x j , is countably additive. With the help of this theorem it is easy to show a sufficient condition for a Gaussian measure μ, defined in the conjugate space Φ' of a countably Hilbert space Φ by a scalar product Β{φ, φ) in Φ, to be countably additive. Let Β{φ, φ) be a scalar product (nondegenerate) in a countably Hilbert space Φ, which is continuous jointly in both arguments relative to the topology of Φ.+ Completing Φ relative to this scalar product, we obtain a Hilbert space Φβ. F r o m the continuity of Β{φ, φ) it follows that the natural imbedding of Φ into Φβ is continuous. Therefore there is an m such that for w > m the imbedding of the Hilbert space Φ^ into Φβ is continuous (Φ^ is the completion of Φ relative to the scalar product (φ, 0 ) J . A sufficient condition for the countable additivity of the Gaussian measure,on Φ' defined by Β{φ, φ) is given by the following theorem. T h e o r e m 1 . In order that the measure μ, defined in the conjugate space Φ' of a countably Hilbert space Φ by a continuous nondegenerate scalar product Β{φ, φ), be countably additive, it is sufficient that for some η the mapping T^^ of Φ^ into Φβ be of H i l b e r t - S c h m i d t class. Proof. T h e scalar product Β{φ, φ) in the Hilbert space Φβ defines a Gaussian measure μ, in the conjugate space Φβ which satisfies the continuity condition. T h e mapping adjoint to m a p s Φβ into Φ^ and is also H i l b e r t - S c h m i d t . N o w apply T h e o r e m 4 of Section 2 to the Hilbert spaces Φβ and Φ^ and to the measure μ, in Φβ. We find that μ, induces a countably additive measure /x^ in Φ^, which in turn induces a countably additive measure in Φ'. Obviously this last measure coincides with the measure μ defined in Φ' by the scalar product Β{φ, φ), which is thus countably additive. T h e proof of T h e o r e m 1 was based upon T h e o r e m 4 of Section 2. T h e central and most difficult point in the proof of the latter theorem was to establish the inequality 1 -

μ{Κ) < c(e +

Η-ηκη

(cf. L e m m a 4 of Section 2). For Gaussian measures one can avoid this inequality by using the more simply proven inequality

t B y T h e o r e m 3 o f C h a p t e r I , S e c t i o n 1 . 2 , it is s u f f i c i e n t t h a t Β{ψ, in e a c h a r g u m e n t s e p a r a t e l y .

φ) b e c o n t i n u o u s

3.2

Gaussian Measures

341

Here C{x, x) denotes a strictly positive-definite quadratic form in the space Tr(C"^) denotes the trace of the matrix C"^, and Ω denotes the region outside the sphere of radius r and center at the origin of i?^. In order to prove inequality (10), we note that

Rn

where χ{χ) denotes the characteristic function of the region ß . Since χ{χ) = 1 for those χ satisfying (Λ:, X) > and vanishes for those χ satisfying (Λ:, Λ:) : ζ r^, then the inequality χ{χ) < (Λ:, x)lr^ holds for aU xe It follows that

(27r)^^r2

{x.x)e-'^^^^^^^dx.

Applying formula (4) of Section 2.2, Chapter I I I , to the right side of this inequality, we obtain

which proves inequality (10). Now we prove the following lemma. L e m m a 2. If μ is the Gaussian measure in a Hilbert space Η which is defined by the scalar product {ψ, φ) in i / , and Γ is a H i l b e r t - S c h m i d t mapping of Η into a Hilbert space H^, then the measure μι in Hi^ induced by Τ and /x, is countably additive. Proof. According to T h e o r e m 2 of Section 2, it suffices, for the proof of the countable additivity of μι, to show that for any € > 0 there is a ball Si{r) with radius r and center at the origin in Hi such that the measure of any cylinder set lying outside Si{r) is less than e. One constructs Si{r) in the following way. L e t T" be the mapping of Hi into Η which is adjoint to Γ , and consider the operator Q = T'T. Since Τ is of Hilbert-Schmidt class, then by T h e o r e m 4 of Chapter I, Section 2.3, 0 is a nuclear operator. L e t r be any number such that T r ρ < er^, where T r Q is the trace of Q, T h e n r is the desired radius. T o prove this assertion it suffices to write out the explicit expression for μι and to apply inequality (10). We omit the details of the argument. T h e o r e m 1 follows directly from L e m m a 2. Indeed, the scalar product Β{ψ, φ) defines a Gaussian measure μβ in the conjugate space Φβ of Φβ. But the mapping of into Φβ is, by hypothesis, of H i l b e r t Schmidt class. Therefore its adjoint Γ^, which maps Φβ into Φ^, is of

342

MEASURES IN

LINEAR TOPOLOGICAL SPACES

Ch.

IV

Hilbert-Schmidt class. But then the measure /x^, induced in Φή by /x^ and Tliy is countably additive by L e m m a 2 , as is therefore the measure μ induced in Φ' by /x^. But μ is none other than the measure defined in Φ' by Β{φ, φ). T h i s proves T h e o r e m 1. In view of T h e o r e m 3 of Section 2 , any Gaussian measure in the conjugate space Φ' of a nuclear space Φ is countably additive (we remark that this assertion can also be proven more simply than the general case, by using inequality ( 1 0 ) ) . Now we show that the requirement of nuclearity of the space is not only a sufficient, but also a necessary condition for every G a u s s i a n measure in the conjugate space Φ' of the countably Hilbert space Φ to be countably additive. T o do this, we need the following estimate for Gau§sian measures in a Euchdean space R^. L e m m a 3. Let μ be the Gaussian measure in n-dimensional Euchdean space defined by the scalar product {x, y) in R^y and let Ω denote the region defined by the inequalities Tr C - 2 V T r C < C\x, x)^TrC

+ 2 VTr

C,

where C{Xy x) is a positive-definite quadratic form in R^^, and T r C is the trace of the matrix C consisting of the coefliicients of the quadratic form. T h e n μ{Ω)>1^^1±~±^\ where λ^,

(11)

are the eigenvalues of C *

Proof. Let χ{χ) denote the characteristic function of the region Ω. Obviously the inequality

c

_ XW

[C{x,x)-TrCY 4

^ 1

is satisfied for all points χ e R„. Therefore

Rn J

Rn Λ

(Γίτ νλν^ -_ 9 {C{x, x)f 2

(12) Tr -(- (Tr i'l'r TV rC(x. CC{x, X) +

Cf

.

* T h e reader c a n w i t h o u t difficulty establish the c o n n e c t i o n o f this l e m m a with t h e well-known C h e b y s h e v inequality of probability theory.

3.2

Gaussian Measures

343

F r o m formula (4) of Section 2.2, Chapter I I I , it follows that

and therefore μ{Ω) > 1 -

I

[(Cix, x)Y - (Tr cy]

άμ{χ).

In order to estimate the integral

we choose a Cartesian coordinate system in R^^^ in which the form C{x, x) reduces to a s u m of squares C{x,x) = X,x\+

...+K=cl

T h e n the above integral assumes the form I (λΛ" 4- ... + λ,,ν^)·^ e x p [ - ^ ( * ? + ... +

^

4)]

But for y # Ä L - J (2π)

e x p [ - i ( * ? + ... + 4)]

= 1,

and 1 (2„).n J

«χρ[-έ(*? + - +

dx = 3,

and therefore the integral under consideration is equal to

Consequently, in view of inequality (11)

= 1 -

λ? + . . . + λΐ 2TrC

T h i s proves L e m m a 3.

344

MEASURES

IN

LINEAR

TOPOLOGICAL

SPACES

Ch.

IV

Consider the Gaussian measure μ in a (real) Hilbert space H, defined by the scalar product (φ, φ) in Η. Let Τ be an operator which maps Η into another Hilbert space H^, and denote by μ^ the measure induced in Hi by μ and T. We prove the following lemma. L e m m a 4. If Τ is not of Hilbert-Schmidt class, and || Γ | | < 1, then for any r > 0 there is a cylinder set Ζ in H^, lying outside the ball Si{r) in H^ with radius r and center at zero, whose measure is greater than 1 /2. Proof. Consider first the case where the positive-definite operator Q = T'T has a pure discrete spectrum. Let λ^, Ag, ... be the eigenvalues of Qy and A^, Ag, ... the corresponding normalized eigenvectors. O b ­ viously the inverse image of the ball 5 i ( r ) under the mapping Τ is the set Ω in Η defined by the inequality ( Γ φ , Τφ) < r^, or equivalently {Qφy φ) ^ r^. In coordinate form the set Ω is defined by the inequality %λ,{φ.Η,γ^/Κ

(13)

A=l We note that the series Σ^=.ι λ^. diverges, because Τ is by hypothesis not of Hilbert-Schmidt class, and so Q is not a nuclear operator. Con­ sequently, there are m and η such that

A,-2V

X

X

X , ^ r \

(14)

We now assert that the desired cyhnder set Ζ in / / is the cylinder set defined in coordinate form by ^

λ,{φ,Η,Υ^τΚ

(15)

A = m 4 1

Indeed, comparing this inequality with (13), we conclude that Ζ hes outside the region ß . Let us now estimate the measure of Z . In view of inequality (14), the set Ζ contains the set Z^, defined by the inequalities

/1-2\//Γ<

X

λ,Χφ, h,Y ^ A-\-2 VA,

where A = ^k=m+i K- Therefore μ{Ζ) L e m m a 3 we have the estimate /'Z

\

^

ί

^' ' -

^m+l

> μ{Ζι).

+ ··· + Xn

2 ( λ . . , + ... + λ„) ·

But

according to

4.1

Fourier T r a n s f o r m s of Measures

345

Since by hypothesis || ΤΊ| < I, it foUows that λ^. < 1 for all k. Therefore we obtain μ{Ζ) ^ μ{Ζ-^ T h u s we have proven that μ{Ζ) ^ ^. M a p p i n g Ζ into H^, we obtain a cyhnder set in H^, lying outside 5Ί(Γ) and having measure at least \ . T h i s proves L e m m a 4 when T'T has a discrete spectrum. When the spectrum of T'T is not purely discrete, the proof is carried out in similar fashion, replacing the vectors h^, Ag, ... by orthonormal vectors φ^, ··· such that (T"Γφ^, ψJ^ > C > 0 (the existence of these vectors follows directly from the fact that the spectrum of T'T is not purely discrete). We omit the details. We can now prove that if a countably Hilbert space Φ is not nuclear, then there exists a Gaussian measure μ in the conjugate space Φ' which is not countably additive. Indeed, since Φ is not nuclear, there exists an m such that the mapping of Φ^ into Φ ^ is not of H i l b e r t - S c h m i d t class for any η > m. T h e n the adjoint operators ( Γ ^ ) ' are also not of H i l b e r t - S c h m i d t class. N o w consider the Gaussian measure μ^^ in Φ ^ defined by the scalar product (F, G)_„,. T h i s measure induces a measure /x^ in each of the Φ^ and a measure μ in Φ'. L e t us show that μ is not countably additive. Indeed, since ( T ^ ) ' is not Hilbert-Schmidt for any n ^ m , then by L e m m a 4 there exists, for any η and r, a cyhnder set Ζ in Φ^ lying outside the ball {FyF)_^ < r^, whose measure is at least \ ( L e m m a 4 applies since it is clear, from the monotonicity of the inner products in a countably Hilbert space, that || || < 1). But then T h e o r e m 2 of Section 2 implies that μ is not countably additive. We have thus proven the following theorem. T h e o r e m 2 . In order that every Gaussian measure in the conjugate space Φ' of a given countably Hilbert space Φ be countably additive, it is necessary and suflScient that Φ be a nuclear space. Obviously the nuclearity of Φ is α fortiori necessary for the countable additivity of all (not only Gaussian) measures in Φ'. 4. F o u r i e r T r a n s f o r m s of M e a s u r e s in L i n e a r T o p o l o g i c a l Spaces

4.1. Definition of the Fourier Transform of a Measure T h e Fourier transform of a nonnegative measure μ in w-dimensional Euclidean space R^, is defined as the function f{x) given by f{x) = ^ e^^-^y^ άμ{γ), Let us carry over this definition to a linear topological space.

(1)

346

MEASURES IN LINEAR

TOPOLOGICAL SPACES

Ch.

IV

Let Φ be a hnear topological space and μ a cylinder set measure in the conjugate space Φ'. We define the Fourier transform of μ as the (non­ linear) functional L{ψ) defined on Φ by L(^) = J.^<^.^)^MF).

(2)

We remark that to compute L{ψ) it suffices to know the measures of half-spaces in Φ'. Indeed, if 9? G Φ, then the inequality ( F , ψ) defines a half-space in Φ', whose measure we denote by μφ{χ). T h e n L{ψ) can be written in the form L ( 9 ) = \e^-dμ,{xy

(3)

We note that for positive λ the half-space ( F , λψ) < χ coincides with the half-space (jP, ψ) ^ xjX, Therefore for all positive λ we have 1(λφ) = J

άμ>^{χ) = \

άμ, (|)

= J

άμ^χ).

(4)

Now if λ < 0, then the half-space ( F , λφ) < χ coincides with the halfspace (F, 9?) > x/Xy and therefore at the points of continuity of the function μλφ{χ) we have

Therefore 1{Χφ) = I e^- άμφ)

=-je^-

άμ, (^)

= J e^'- άμφ{χ)

also holds for λ < 0. It is easy to show that if 7^ is a finite-dimensional subspace in Φ and μψ is the measure in the factor space Φ'ΙΨ^ corresponding to /x, then for any φβΨ one has 1{φ)=(

/^^^^^

άμψ(Ρ),

(5)

Indeed, if φ G Ψ, then the half-space ( F , φ) ^ χ consists of cosets with respect to the subspace in Φ'. Therefore the /it-measure of this halfspace in Φ' coincides with the /x,p-measure of the half-space ( F , φ) ^ x in Φ'ΙΨ^. Since the Fourier transform of a measure is uniquely defined by the measures of half-spaces, this proves (5).

4.2

Fourier T r a n s f o r m s of M e a s u r e s

347

Äis an example, let us calculate the Fourier transform of the G a u s s i a n measure μ defined by a functional ß(
r«

1

/

-x'

the functional Ε{φ) is given by 1

r"^

/

X'^

\

But this integral equals ^ - ^ ^ ^ ^ Μ Γ ^ It follows that 1{ψ)

=

^-i^iv.^).

(6)

4.2. Positive-Definite Functionals on Linear Topological Spaces L e t L ( 9 ) be a functional on a hnear topological space Φ. T h i s functional is called positive-definite if V

L(9>, - ψ,)ξ-ξ^ > 0

(7)

for any elements φι, in Φ and any complex n u m b e r s ^i, i^. An example of a positive-definite functional is furnished by any functional Ε{ψ) which is the Fourier transform of a cylinder set measure in the conjugate space Φ' of Φ (recall that we are considering only positive normalized measures). Indeed, suppose that Ε{φ) is the Fourier transform of a measure μ. L e t Ψ be the finite-dimensional subspace spanned by the elements φι, φ,^^ and let μψ be the measure in the factor space φ'/ψ^ corresponding to the measure μ. T h e n for φ sW, ^{ψ) is given by formula (5). But then we have % L(φ, - φ , ) ί , | , =

^

a ,

ί m

/''^'^rn' 2

from which it is evident that ^ φ ) is positive-definite.

dμψ{F)

348

MEASURES

IN LINEAR

TOPOLOGICAL

SPACES

Ch. IV

Let us remark that if μ satisfies the continuity condition, then its Fourier transform is continuous. In fact, suppose that the sequence { 9 ^ } converges to the element e Φ. L e t /x^ be the measure on the line corresponding to the element φ^, and /x^ the measure corresponding to φQ. T h e n L{9n)=

i^^^^i^nW,

«=

0,1,....

J

But the continuity condition says that J / ( ^ ) dH'nix) = jf{x) αμο{χ) for any bounded continuous function f{x). L e t t i n g / ( j c ) = e^^, we obtain lim^^oo L{Tn) ^ ^(TO)> which shows that ^φ) is continuous. Lastly, we note that L ( 0 ) = 1, since the measure / X Q corresponding to the zero element is concentrated at the point χ ^ 0 , and therefore e-^^oW

1.

T h u s we see that the Fourier transform ^φ) of any measure on the cyhnder sets in Φ' is positive-definite and continuous (in the sequential sense) and L ( 0 ) = I. N o w we show that these conditions are not only necessary, but also sufficient for a functional L((p) to be the Fourier transform of some cyhnder set measure in Φ'. T h e o r e m 1 . In order that a functional L{φ) on a hnear topological space Φ be the Fourier transform of some cylinder set measure in the conjugate space Φ', it is necessary and sufficient that L{φ) be positivedefinite and continuous (in the sequential sense) and that L ( 0 ) = 1 . Proof. T h e necessity of the conditions was proven above. T o show their sufficiency, let L((p) be a functional satisfying the conditions of the theorem. Considering Ε{φ) on a finite-dimensional subspace Ψ of Φ, we obtain a positive-definite continuous function Lψ{φ) on Ψ. By Bochner's theorem (cf. Chapter I I , Section 3 . 2 ) this function is the Fourier transform of a positive measure μψ defined in the conjugate space Ψ' of Ψ. But we have seen in Section 3 . 1 that Ψ' can be identified in a natural way v/ith the factor space Φ'/Ψ^, where Ψ^ consists of all hnear functionals F which vanish on Ψ, It follows that in each of the factor spaces Φ'/Ψ^, where Ψ is finite dimensional, there is defined a measure μψ. It remains for us to show that these measures are compatible and satisfy the continuity condition.

4.2

Fourier T r a n s f o r m s of Measures

349

T o prove compatibility, consider two finite-dimensional subspaces Ψι, with Ψι C ψ^ in Φ, and let and μ^ be the measures correspond­ ing to them. We have to prove that μ^ coincides with the measure ν induced in the factor space Φ'/Ψ^ by JLIG- I^i other words, we must prove that μι{Α) = v{A) = μ,[ρ-\Α)]

(8)

for any set A in Φ'/Ψ^, where Q denotes the mapping of Φ'ΙΨΐ onto Φ'ΙΨΙ by which the coset F + Ψΐ is carried into the coset F + Ψ^. We prove (8) by showing that the Fourier transforms of the measures μι and V coincide. T h e Fourier transform of μι is by definition of μι the function Lι{φ) defined on Ψι and coinciding there with T h e Fourier transform of V is also defined on Ψι, and is given by (9)

dv(F).

ei(F,^)

If φ G Ψ I, then the value of {F, ψ) is the same for all functionals F belonging to the same coset FQ + Ψ\, Since, moreover, ν and μ^ are related by (8), we can rewrite (9) in the form

T h u s (9) is the Fourier transform of μ^ for all elements φ G Ψι. But by definition of μ^ this Fourier transform is that function L^^) on Ψ^ which coincides there with L{(p). But L^^) and L^^) coincide on Ψι, since L^ψ) = Ε{φ) = L2{φ) for φ e Ψι. T h u s the Fourier transforms of μι and ν coincide. But then μι and ν coincide, which proves that the measures μ ψ are compatible. We can thus associate with the functional ^φ) a cylinder set measure μ in Φ'. It remains for us to show that μ satisfies the continuity condition. For this we use the following theorem from the theory of Fourier in­ tegrals: // a sequence

{/ut^}

of positive normalized measures is such that lim ί e'^^ άμη{χ) = ί β^'- dμo{x\ W->00 J

J

for any value of X, then the measures μ^ converge weakly to μ^.^ ^ A sequence

o f m e a s u r e s is s a i d t o c o n v e r g e H m ί / ( Λ ; ) άμη{χ) Η-^» J

for any b o u n d e d

continuous function

f{x).

=

(f{x) J

w e a k l y t o a m e a s u r e μο, άμ,{χ)

if

350

MEASURES IN L I N E A R TOPOLOGICAL

SPACES

Ch.

IV

T o prove, nov^, that μ satisfies the continuity condition, let {φ^} be a sequence of elements in Φ which converges to an element φο» /x^ be the measure corresponding to φ^, w = 0 , 1, ... . T h e n for any real value of λ we have 1(λφη)

=

j e''^ άμη{χ\

n =

0,

1,...

(cf. formula ( 4 ) ) . Since in view of the continuity of Ε{φ) we have lim^^oo ^^Ψn)

= H^9o)y then for any λ lim|e*-^-^,(^)= |.^^-φο(Λ

But this, as we said, imphes that the measures / L C ^ , η = 1, 2 , ... converge weakly to μQ. T h u s the measure μ which we have constructed on the cylinder sets of Φ' satisfies the continuity condition (cf. Section 1 . 4 ) , which concludes the proof of the theorem. If Φ is a nuclear space, then by T h e o r e m 3 of Section 2 , any cylinder set measure in Φ' which satisfies the continuity condition is countably additive. We therefore have the following assertion. T h e o r e m 2 . Any continuous^ positive-definite functional Ζ,(φ) on a nuclear space Φ, such that L ( 0 ) = 1, is the Fourier transform of a countably additive positive normalized measure in Φ'. T h i s theorem is simply Bochner's theorem for nuclear spaces.

5. Q u a s i - I n v a r i a n t Measures in L i n e a r Topological Spaces 5 . 1 . Invariant and Quasi-Invariant Measures in Finite-Dimensional Spaces In this section we will consider questions connected with the trans­ formation of measures in linear topological spaces by parallel displace­ ment. By the term **measure" we mean a positive countably additive measure μ{Χ) on the Borel sets in a conjugate space Φ', which is regular in the sense of Caratheodory, and such that the entire space is a countable union of sets of finite measure (this last property is called the σ-finiteness

of μ). t S i n c e a n u c l e a r s p a c e h a s a m e t r i c t o p o l o g y (cf. f o o t n o t e o n p . 57), sequential continuity are equivalent.

continuity and

5.1

Quasi-Invariant Measures

351

We start by considering measures in finite-dimensional linear spaces. In a finite-dimensional linear space there exists L e b e s g u e measure μο{Χ) v^hich is invariant under any parallel displacement in Ä^. In other words, the measure μο{Χ) is such that μο{Χ) = μο{γ + Χ) for all vectors y and measurable sets X in 7?^^. T h e property of invariance under parallel displacement is characteristic for L e b e s g u e m e a s u r e s — any two measures which are invariant under all parallel displacements are identical up to a constant factor. Let us now consider measures which are equivalent to L e b e s g u e measure. We say that two measures μ and ν are equivalent, if they have the same family of null sets (i.e., if μ{Χ) = 0 implies v{X) = 0 and conversely). A description of all measures equivalent to L e b e s g u e measure is given by the following theorem. T h e o r e m 1 . Any measure μ in i ? ^ which is equivalent to L e b e s g u e measure has the form μ{Χ)=

ί

f{x)dx,

where f{x) is a strictly positive function which is s u m m a b l e over every bounded set in Proof. Since μο{Χ) = 0 imphes μ{Χ) = 0, by the R a d o n - N i k o d y m theorem^ there exists a finite-valued nonnegative measurable function f{x) such that μ{Χ)=

( f{x)dx

(1)

for all measurable sets X in 7?^. L e t XQ denote the set of points at which f{x) = 0. Obviously μ{Χ,)=

ί

f{x)dx

= 0;

since μ and /XQ are by hypothesis equivalent, μο{Χ) = 0. Consequently f{x) is almost everywhere positive. Since f(x) can be altered on a set of ^ T h e R a d o n - N i k o d y m theorem says the

following:

S u p p o s e t h a t μ a n d ν a r e m e a s u r e s s u c h t h a t μ{Χ) Then

there exists a

finite-valued

nonnegative MAT)

f o r a l l m e a s u r a b l e s e t s X.

=

J ' X

T h e f u n c t i o n f{x)

= 0 f o r a l l s e t s Xof

m e a s u r a b l e f u n c t i o n f{x)

v-measure zero. such

that

f{x)dv(x)

is d e f i n e d u p t o a s e t o f v - m e a s u r e z e r o .

352

MEASURES

IN

LINEAR

TOPOLOGICAL

SPACES

Ch.

IV

L e b e s g u e measure zero without affecting (1), we can suppose that it is positive everywhere. Lastly, we show that f{x) is summable over any bounded set. Since μ and μ^ are equivalent and the L e b e s g u e measure of a point x Ε R^^ is zero, then μ{{χ}) — 0 for all χ G R^^. But since we assumed that μ is regular in the sense of Caratheodory, for every χ e R^ there is an open set V{x) containing χ whose /x-measure is finite. Since any closed bounded set X in R^^ can be covered by a finite number of the V{x) (and hence any bounded set can), the /x-measure of any bounded set is finite. We remark that the L e b e s g u e measure /XQ is expressible in terms of μ by the formula

UX)

=

j

άμ{χΙ

As a matter of fact, if μ and ν are measures equivalent to L e b e s g u e measure, then they are mutually equivalent, and considerations similar to those used in the proof of the preceding theorem show that we have

.(X)=

/,Χχ)αμ{χ\

where f,iv{x) is summable (relative to μ) over every bounded set and is positive for all x. Measures μ which are equivalent to L e b e s g u e measure have the following weakened property of invariance under parallel displacement. If a set X has /x-measure zero, then every translate of X has/x-measure zero. Indeed, from the invariance of L e b e s g u e measure and the definition of equivalence we have the chain of implications ^(X) ^ 0 -

μ,{Χ)

= 0 -

μ,{γ + X) = 0^

μ{γ + Χ) =

0.

A measure which has the property that μ{Χ) = 0 implies μ{γ + X) = 0 for all y will be called quasi-invariant (relative to parallel displace­ ment = translation). We have therefore proven that all measures which are equivalent to L e b e s g u e measure are quasi-invariant. T h e converse is also true. T h e o r e m 2. If a measure μ is quasi-invariant, then it is equivalent to L e b e s g u e measure. First we prove the following lemma.

5.1

Quasi-Invariant Measures

353

L e m m a 1 . If a measure μ is quasi-invariant, then the /x-measure of any bounded set is finite. Proof. T h e quasi-invariance of μ implies that the /x-measure of each point xe is zero. Indeed, if for some XQ one had μ({χο}) > 0, then this would imply μ{{χ}) > 0 for all x. T h u s any set containing a nondenumerable number of points would contain an infinite number of points whose measures all exceed some fixed positive constant, and therefore the set would have infinite measure. But this clearly contradicts the σ-finiteness of μ. Now the regularity of μ implies that for every Λ: G there is an open set V{x) containing χ which has finite /x-measure. Since any closed bounded set (and therefore any bounded set) can be covered by finitely many of the V{x)y the /x-measure of such a set is finite. Proof of Theorem. S u p p o s e that μ is quasi-invariant. Clearly if X is bounded, then μ{γ + X) is finite for all y. As for its measurability as a function of jy, this follows at once from μ{γ + Χ)=

Γ ·'

χ{χ-γ)άμ{χ),

(2)

-00

where χ is the characteristic function of X. Further, if F is a bounded set, then we can obviously find a bounded measurable set Ζ such that {y + X)CZ for all yeY; hence μ{y + X) ^ μ{Ζ) < oo for all yeY. N o w suppose that μ{Χ) > 0, μο{Χ) = Ο, and μ^{Υ) > 0. T h e n by the quasi-invariance of μ we have μ{y -\- X) > 0 for all y. T h e n 0 <

ί μ{y + X)dy=

ί

\Γ

χ{x-y)dμ{x)\dy x{x-y)dy

dμ{x\

γ

where Fubini*s theorem is applicable because our definition of quasiinvariance includes the assumption of σ-finiteness. But ί χ{χ -y)dy

= μο{Υη{χ

- Χ)) < μο{χ -Χ)

= 0

(4)

J γ

for every Λ:, since μQ{x — Χ) = μο{Χ) = 0. But then the right side of (3) vanishes, which is a contradiction. T h u s , we have shown that any bounded set^ of L e b e s g u e measure t T h e u s e of b o u n d e d sets a n d L e m m a 1 in o r d e r to avoid the c o n s i d e r a t i o n of

sets

h a v i n g (possibly) infinite / i - m e a s u r e a p p e a r s u n n e c e s s a r y , as F u b i n i ' s t h e o r e m for n o n n e g a t i v e f u n c t i o n s is v a l i d w i t h o u t a n y s u m m a b i l i t y r e s t r i c t i o n s .

354

MEASURES IN LINEAR TOPOLOGICAL SPACES

Ch.

IV

zero has /x-measure zero. Since any set of L e b e s g u e measure zero can be written as a countable union of bounded sets of L e b e s g u e measure zero, then μ{Χ) = 0 for all sets X of L e b e s g u e measure zero. T o show the converse, suppose that μο{Χο) > 0 but μ{Χο) = 0. L e t {xj^} be an everywhere dense sequence of points in R^, and define Ζ as the union Z=

y^{x,

It is not hard to show+ that /Xo(/?n -

+ Xo).

(5)

Z) = /^o(^i) = 0· But

μ{Κ,) = μ{Ζ) + μ{Ζι) < μ{Ζι) + V μ{χ, + ^ο).

(6)

Since μ{Χο) = Ο and μ is quasi-invariant, then μ{χ^ + ^ο) = Ο for all k. Furthermore, as we saw above, / X o ( ^ i ) = 0 imphes μ{Ζι) = 0. Therefore μ{Rn) = 0. T h u s μ is either equivalent to L e b e s g u e measure or else vanishes identically. We have thus proven that the class of quasi-invariant measures in Rn coincides with the class of measures equivalent to L e b e s g u e measure. Therefore, all quasi-invariant measures in are equivalent to one another.

5.2.

Quasi-Invariant Measures in Linear Topological Spaces

Let us now consider measures in infinite-dimensional linear topologi­ cal spaces. T h e definition of a quasi-invariant measure can be carried over formally to this case, by calling a measure in a linear topological space quasi-invariant if parallel displacement takes sets of measure zero into sets of measure zero. However, this formal extension is un­ successful, owing to the fact that for the most important classes of infinite-dimensional spaces there are no nonzero measures which are quasi-invariant in the sense indicated. • t I n d e e d , i f μο{Χο) € >

0, a s e q u e n c e

>

0, then b y the t h e o r y of differentiation w e c a n

MnXo) Let ^

/ =

and /' be {Xi +

find,

any

fixed

cubes such that /

> (1"€)μο(Λ). lies in t h e interior o f / ' . C l e a r l y the

family

7*} o f c u b e s c o v e r s / i n t h e s e n s e o f V i t a l i , a n d s o b y V i t a l i ' s c o v e r i n g t h e o r e m

t h e r e e x i s t s a s e q u e n c e { / * } o f d i s j o i n t c u b e s f r o m JJf s u c h t h a t μο(Ι—Uj^.j/*) a n d i n a d d i t i o n t h e /jt c a n b e c h o s e n s o t h a t t h e y l i e i n €μο(Ι%

for

of c u b e s w h o s e d i a m e t e r s tend to zero a n d s u c h that

where Ζ

i s d e f i n e d b y ( 5 ) . S i n c e c i s a r b i t r a r y , μο(Ι — Ζ ) =

a n a r b i t r a r y c u b e , it f o l l o w s t h a t μο(Ηη — Ζ ) =

0.

=

B u t t h i s i m p l i e s t h a t μ^(Ι — Ζ ) Ο, a n d e i n c e /

0, < is

5.2

Quasi-Invariant M e a s u r e s

355

L e t us consider measures μ in the conjugate space Φ' of a countably normed space Φ. We shov^ that if the spaces Φη^ηά Φ^^ι are different for every n, where Φ = Π^^χ Φ^, then there exists no quasi-invariant measure in Φ'. Indeed, Φ' is the union of the subspaces Φ^ conjugate to the Φ^, Φ' υΓ=ι Φή, where Φ^ C Φ^ C .... Therefore

=

n-*Qo

as the Φ'η are Borel sets in Φ' for which /x is therefore defined (cf. the opening remarks in Section 2.2). Since by hypothesis μ{Φ') Φ Ο, there is an η such that μ(Φ^) Φ 0. Since Φ^ and Φ^^.^ are diflFerent from one another, then Φ^ and Φ' differ from each other. We decompose Φ' into cosets with respect to Φ^. E a c h of these cosets is obtained by a parallel displacement of Φ^, which has nonzero measure, and so by the quasiinvariance of μ these cosets have nonzero measures. Since the family of all these cosets has the power of the continuum, we arrive at a contra­ diction with the σ-finiteness of /x. T h i s assertion holds also for the conjugate space Φ' of any normed space having a countable everywhere dense set. I n particular, there exists no quasi-invariant measure in a Hilbert space (cf. Section 5.3). As in many similar cases, the difficulties which arise are successfully overcome by considering rigged Hilbert spaces. T h u s , let Φ€ HC Φ' be a rigged Hilbert space, i.e., a nuclear space Φ in which there is given a scalar product ( φ , φ) (as in Section 4 of Chapter I , Η denotes the completion of Φ in the norm || φ || = V ( φ , φ)). With each element φ βΦ we associate a functional Fy, on Φ, defined by {Fy,, φ) = ( φ , φ). W e obtain thereby an (antilinear) imbedding φ'-^Fy, of Φ into Φ'. It is obvious that the functionals of the form Fy,, φ βΦ, are everywhere dense in Φ'. We will say that a measure μ in Φ' is quasi-invariant, if μ{Εγ + X) =0 for every element φΕΦ and every set X such that μ{Χ) = 0. T h u s , we eliminate the requirement that every translation carries sets of measure zero into sets of measure zero, requiring that this be true only for translations by the elements Fy,, φβΦ, We remark that since the elements of the form Fy, are everywhere dense in Φ', then the translations by elements F are ^'sufficiently numerous.'* T h i s means that if Φ'ΙΨ^ = i? is a finite-dimensional factor space, then any translation in R can be induced by a translation in Φ' corresponding to some element Fy,, where φθΦ. T h i s assertion follows from the fact that the map of Φ into R is everywhere dense in R, which, in view of the finite dimensionality of R, means that it coincides with R,

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We w i n prove that there exist quasi-invariant measures in the conjugate spaces of nuclear spaces (here, of course, quasi-invariance is understood in the sense j u s t indicated). T h a t is, we will show that if an imbedding of Φ into Φ' is defined by a continuous Hermitean functional Β{ψ, φ), then the Gaussian measure defined by this functional is quasi-invariant. In other words, we will prove the following theorem. T h e o r e m 3. L e t Β{φ, φ) be a nondegenerate positive-definite Hermitean functional, continuous in each argument, on a nuclear space Φ, and let μ be the Gaussian measure in Φ' defined by 5(
^ ^ ^ ^ ^ < .«"^"».

Proof.

(7)

By the definition of a Gaussian measure we have^ m(2)

- ( i r /

exp[-ifi^(/",i^)]

dj-

and μ{Ρ,, + Ζ ) = ^ ί (2π)" 1

where B\y{P^,

txpl-^B^P,Ρ)]

d^P

exp[-^B^{P-P^,P-P„)]d^P,

is the functional on φ ' / ψ « defined by ^ ( 9 5 , φ), and

* F o r b r e v i t y o f n o t a t i o n w e d e n o t e b y Ρ t h e c o s e t w i t h r e s p e c t t o Ψ" c o n t a i n i n g t h e e l e m e n t F e Φ', a n d hy Ρ,^ + A t h e p r o j e c t i o n i n t o ΦΊΨ'> o f t h e c y l i n d e r s e t Fy, A- Z.

5.2

Quasi-Invariant Measures

357

άψΡ is the L e b e s g u e measure in Φ'ΙΨ^ corresponding to the scalar product Βψ{Ρι,Ρ^, Therefore

= sup exp[B^(F,^^) - ^B^(P^,F^)]

< sup

cxp[B^{P,P^)]

F r o m the definition of the functional J5,p(J^i,/^a) (^f- Section 3.2) it follows that Β^{ί^,Ρ,)

= {Ρ,φ)

for any Ρ in Φ'/Ψ^^ where ( F , 0 ) denotes the value of the functional F for the element φ. Ii Ρ e Ay then by hypothesis one can choose F inP so t h a t | | F | U < Then \B^PyP^)\

<\{Ρ,φ)\

^\\Ρ\\_η\\φ\\η^Κ\\φ\\η.

Therefore /x(Z) which proves the lemma. Proof of the Theorem. L e t AT be a set of /x-measure zero, and let φ be any element in Φ . We have to show that the s e t F ^ + Xy the translate of X by the element Fy,, also has zero /x-measure. T o do this, it suffices to show that /x(Fy, + X) < € for any e > 0. T h u s , let € > 0. Since /x is countably additive and Φ ' is the union of balls < / ? , one can find η and R such that / χ ( Φ ' — S ) < J c , where S is the ball || F ||_^ < Ry and F^, e Φ ; . L e t X^ = X η S^y where 5 i is the ball \\F\\_^ < / ? + and X^ = X - X^. Obviously the set Fy, + X^ hes in the complement of S , and therefore μ{Fy, + X^) < i c . Let us show that also μ{Ρ^, + X^ < ^e. Indeed, since /χ(ΑΊ) = 0, it can be covered by a countable union U*^i of cylinder sets such that /x(Z,) <

e x p [ - ( Ä + II

lUJII φ

u

Since X^ C 5 , the cylinder sets Z^, 1 < Λ < oo, can be chosen so that their bases Aj, lie respectively in the projection of the ball in Φ ' / Ψ ^

358

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{Ψ^ denotes the generating subspace of Zj.). We may suppose without loss of generality that φ eWj^ for all k. Since the Aj^ lie in the projection of the ball = {|| F\\_^ < / ? + || ||_ then by L e m m a 2 μ(Ρ^ + Ζ , ) < exp[(Ä + II F , II.Jll φ |U]^(Z,) and therefore

A;=l

< exp[(Ä +

|1_„)|| φ ||„]

μ(Ζ,) <

fc=l

T h u s ^{Fy, + Z i ) < J e . But this means that μ{Ρ^ + X) = μ{Ρ^ + Χ,) + M F , +

^2)

< e,

which proves the theorem. T h u s , Gaussian measures in the conjugate space Φ' of a nuclear space Φ are quasi-invariant. One can say that to every continuous positivedefinite nondegenerate Hermitean functional on Φ there corresponds a quasi-invariant measure. If Φ is infinite dimensional, then there exist infinitely many pairwise inequivalent quasi-invariant measures in Φ' (recall that in a finite-dimensional space all quasi-invariant measures are equivalent to one another). In fact, let β ι ( φ , φ) and Β2{φ\ φ) be positive-definite continuous nondegenerate Hermitean functionals on a nuclear space Φ, and let μ^ and ^ 2 be the Gaussian measures in Φ' defined by these functionals. A s s u m e that ^ 2 ( 9 » Φ) is bounded relative to the scalar product defined by Βι{φ,φ), i.e., that there exists Μ > 0 such that the inequahty \Β^{φ,ψ)\

<Μ|5ι(φ,φ)|

£2(9?,

holds for all ψ ΕΦ. T h e n φ) defines a positive-definite bounded linear operator A in the space Η (the completion of Φ relative to the norm || φ || = Λ/βι(φ, φ)) defined by Β,{Αφ,φ)

= Β2{ψ.φ).

(8)

T h e following assertion holds: If the operator A defined by ( 8 ^ has a discrete spectrum, and the series Σ^=ι consisting of the eigenvalues of A converges, then μ-^ and μ^ are inequivalent. T h e proof of this assertion is based upon L e m m a s 2 and 3 of Section 3. Namely, consider the ball S in Φ defined by 5 ι ( φ , ψ) < i ? ^ and denote the map of S in Φ' by 5 ' . It is easy to show, using L e m m a 3

5.3

Quasi-Invariant M e a s u r e s

359

of Section 3, that μι{3') = 0. At the s a m e time, if Σ^^ι λ^. converges, then using the estimates from L e m m a 2 of Section 3, one can show that for R sufficiently large /X2(5") Φ 0. T h i s shows that and μ2 are inequivalent.^ Using these statements, it is not difficult to construct an infinite set of pairwise inequivalent quasi-invariant measures in Φ'. T o do this, it suffices to consider an infinite sequence Βι{φ, φ), -82(93, ^ ) , ... of positive-definite Hermitean functionals on Φ such that the operator An, defined by Β^,{Α^φ, φ) = Βη^ι(ψ, ΦΙ has a discrete spectrum, and the series consisting of its eigenvalues converges. It would be very interesting to give a complete description of all quasi-invariant measures in nuclear spaces. 5.3. Quasi-Invariant Measures in Complete Metric Spaces

In the previous paragraph it was shown that under very general condi­ tions there exist no measures in the conjugate spaces of countably normed spaces which are quasi-invariant relative to all translations. We now prove a similar result for complete linear metric spaces. T h e o r e m 4. L e t yl be a complete metric linear space containing a countable everywhere dense set, and such that the absolutely convex hull of any compact set* Ζ in is nowhere dense in A, T h e n the only quasi-invariant (under all translations) measure on A is the identically zero measure. Proof. First we show that if there is no normalized quasi-invariant measure (i.e., a quasi-invariant measure such that μ{Λ) = 1), then the only quasi-invariant measure in A is identically zero. Indeed, let /Ll be a quasi-invariant measure in A, Since μ is σ-finite (recall that we are only considering such measures), A can be written at a countable union of disjoint sets A^, A^y ... having finite positive /x-measure. L e t f{x) be defined on A by f{x) = l|2^μ{Λf.) if χ e and set iX)=\

Αχ)αμ(χ).

" O n e c a n s h o w t h a t i f t h e p r o d u c t n*.i

(9)

Κ c o n v e r g e s , t h e n μι a n d μζ a r e e q u i v a l e n t .

I n t h i s c a s e A is t h e s u m o f t h e i d e n t i t y o p e r a t o r a n d a n u c l e a r o p e r a t o r . * B y the absolutely convex hull of a set X combinations

of the f o r m

OiXi

+

... +

we m e a n the set

OnXnt w h e r e x^eX,

LOLL + ... + ION I < 1.

c o n s i s t i n g o f all l i n e a r 1 <

Ä <

n,

and

360

MEASURES IN

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Since ,,ίΑΛ — V ^ =_ Κ/1) —= V |)ΚΛ-) = Χ „iA\

2k

1,

A=l ^

k=l

then J/ is a normahzed measure in Λ, F r o m the quasi-invariance of μ it follows easily that ν is also quasi-invariant. But by hypothesis there is no quasi-invariant normalized measure on / I . Consequently μ{Λ) = 0 . T h u s , our problem has been reduced to proving that there are no normalized quasi-invariant measures in Λ. S u p p o s e that μ is a countably additive measure in Λ such that μ{Λ) = 1. We show that for any η there is a compact set in Λ such that μ{Χη) ^ 1 — Indeed, choose a countable everywhere dense set Λ:2, ... in yl and consider the closed balls 5^.^ with centers at Xj. and radii p~^. Since for any fixed p the balls S^j,, ^ 2 ^ , ... cover Λ (since the set {xj^} is everywhere dense), in view of the countable additivity of μ there is a number k{p) such that the measure of the set

is not less than 1 — (l/2^w). L e t = CQ^i Xnp'y we show that is the desired set. Indeed, it is obvious that μ{Λ-Χ„)^^μ{Λ-Χ„,)^%^

= ΐ,

X^

(10)

from which it follows that μ{Χη) ^ 1 — w"^. Further, for any p the set Xγ^ is covered by the finite set of balls 5ip, of radius p~^. Finally, is closed because each of the Sj^^ is closed and consequently Χγ^ρ, as the union of finitely many of the Sj^pj is closed. But in a complete metric space any closed set Ζ which can be covered by a finite number of balls of any preassigned radius is compact.^ Consequently X^^ is compact. Now let X = U^^i X^. F r o m the relation μ{Χη) ^ 1 — it follows that μ{Χ) = 1. But then the (nonclosed) hnear span Jt of X has measure 1.^ + Let us show that the set ^ does not coincide with the ^ T h e p r o o f o f t h i s a s s e r t i o n is c a r r i e d o u t i n t h e s a m e w a y a s t h e p r o o f o f t h e c o m p a c t ­ ness of a closed b o u n d e d set in a

finite-dimensional

space, with the sole difference that

the coverings of Ζ by balls of arbitrarily small radius play the role of the partitions. ^ B y t h e l i n e a r s p a n A O f a s e t X w e m e a n t h e s e t c o n s i s t i n g o f a l l finite l i n e a r c o m b i n a ­ tions

λγΧι

4-

...

-I-

KXfi

of elements of

X.

t W e r e m a r k t h a t t h e q u e s t i o n o f t h e m e a s u r a b i l i t y o r n o n m e a s u r a b i l i t y o f X is i n c o n ­ s e q u e n t i a l to the proof, a n d c a n b e a v o i d e d in v a r i o u s w a y s , o f w h i c h p e r h a p s the s i m p l e s t is t o o b s e r v e t h a t in t h e v e r y l a s t s t e p o f t h e p r o o f X c a n b e r e p l a c e d b y

X.

5.3

Quasi-Invariant Measures

361

entire space Λ. T o do this, consider the compact sets = (J^^i Xj,, and let denote the absolutely convex hull of 7^. By the conditions of the theorem, the sets are nowhere dense in Λ, Therefore the sets kY^^ consisting of all elements of the form ky^ y e are also nowhere dense in Λ. Since a complete metric space cannot be written as a count­ able union of nowhere dense sets (cf. Chapter I, Section 1.1), the union Y =r- Uk=i Un^ikYn does not coincide with Λ, But X C y , since if xeJt, then χ = X-^x^ + ... + X^Xpy where x^eX^^j^^ and therefore X e P ^ , where η = maxi^^-^p n{j) and k > Σ ^ ^ ι | λ^· |. T h i s proves that the set Jty having measure 1, does not coincide with Λ. Now let y be any element in A which does not lie in Jt, Since is hnear, then Ä and j ; + ^ are disjoint. Since therefore y + ^ lies in the complement of Jt and μ{^ί) = 1, then μ{y + -^) = 0. It foUows that μ is not quasi-invariant. We now show that it follows from this theorem that there exist no quasi-invariant measures in infinite-dimensional complete normed spaces having countable everywhere dense sets. In fact, we have to show only that in such spaces the absolutely convex huU of any compact set is nowhere dense. L e t be a compact set in A^ and S{xQy r) the ball in Λ with radius r and center at XQ. Since X is compact, it can be covered by a finite number of balls S{xj.y ^ r ) , 1 < Ä < w, of radius ^r. Therefore its absolutely convex huU X lies in the absolutely convex hull S of the set S = U^^i S{xj^y i r ) . Any element y e S can be represented in the form y =

λ ι ( ^ ι + >ί)

+

... + λ„(Λ:^ +

y^l

where | | + ... + | | ^ 1 and || I K ^r, ζ = 1, w. But aU elements X^Xi + ·.· + K^n the subspace V spanned by x-^, jc^, and II Aiji + ... + Xnyn II < H\ λ ι I + ... + I λ , I) < i r . Therefore every element in S can be written in the form y = ν -\- Zy where ve V and || ;2r || < ^r. T o show that the closure of S does not contain the ball S{XQ, r), it suffices to find an element y^ 6 S ( 0 , r) which cannot be represented in the f o r m a l = ν — XQ + z, where ve V and || «s: || < ^r. But the existence of such an element follows directly from the finite dimensionality of V and the infinite dimensionality of AA By T h e o r e m 4 this imphes the + T h i s r e s u l t is u s u a l l y s t a t e d a s a t h e o r e m for t h e c a s e r =

1, f r o m w h i c h t h e r e s u l t

for a r b i t r a r y r > 0 is a trivial c o n s e q u e n c e . T h e s t a t e m e n t is t h e f o l l o w i n g . I f F is a c l o s e d s u b s p a c e o f a n o r m e d l i n e a r s p a c e Λ a n d Y ^ Λ, t h e n f o r a n y € > 0 t h e r e e x i s t s a n e l e m e n t xE

Λ s u c h t h a t || Λ; || =

F i s t h e l i n e a r s p a c e s p a n n e d b y JCQ,

1 a n d || Xn-

— x\\

>

1 — c for all

6 F. I n o u r

case

362

MEASURES IN

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nonexistence in Λ of quasi-invariant measures. In the same way one can prove the nonexistence of quasi-invariant measures in any complete countably normed space having a countable everywhere dense set.

5.4. Nuclear Lie Groups and Their Unitary Representations. The Commutation Relations of the Quantum Theory of Fields T h e quasi-invariant measures which we have constructed in Section 5 . 2 find applications in the theory of infinite-dimensional L i e groups. Let G be some (topological) group. We will call G a nuclear Lie group, if there exists a neighborhood of the unit element in G which is homeomorphic to a neighborhood of zero in a countably Hilbert nuclear space Φ. As a rule, nuclear L i e groups are considered for which Φ is a rigged Hilbert space, i.e., such that a scalar product {φ, φ) is defined in Φ. Every nuclear space Φ can be looked upon as a commutative nuclear L i e group. L e t us present a somewhat more comphcated example of a nuclear L i e group. S u p p o s e that Φ€ HC Φ' is a rigged Hilbert space. T h e elements of the group GQ will be all triples g = (φ, 0 ; α), where φ and φ are elements of Φ, and α is a complex number of unit modulus. We introduce a multiplication in GQ, setting glg2

=

{ψΐ> Φΐ\

=

{ψι

+

^ΐ){ψ2^

Φ2\ « 2 )

Ψ2> Φι +

^^^^

0 2 ; e^^'^i^yOociOL^)

((φ, φ) is the scalar product in Φ). T h i s group is connected with the commutation relations of the quan­ tum theory of fields. In quantum mechanics a system having one degree of freedom is studied by means of operators p and q which are connected by the commutation relation pq-qp=

1.

T h i s commutation relation is the commutation relation for the operators of the L i e group G whose elements are triples of numbers {x, y, cx), α 7^ 0, and multiphcation is defined by (Λ^Ι,

Jl,

αι)(Λ^2»

«2) =

(Xi

+ x^, yi +

3^2.

e^^avia^ag).

(12)

In the same way, the consideration of a system with η degrees of freedom leads to the system of commutation relations

5.4

Quasi-Invariant Measures

363

T h e s e are the commutation relations for the operators of the L i e group G whose elements have the form (jc, y, a ) , where χ and y are vectors in n-dimensional space, and multiphcation is defined by (12), the sole diflFerence being that instead of x^yi one has to take the scalar product (^2, >Ί)· Finally, the consideration of quantized fields (systems with an infinite number of degrees of freedom) leads to an infinite system of commutation relations of the form (13). It is natural to regard these relations as the commutation relations of the nuclear L i e group GQ. We will consider here unitary representations of the g r o u p s Φ and GQ. By a unitary representation of any group G we mean a continuous operator-valued function U{g) defined on G , whose values are unitary operators in a Hilbert space i), such that

for any two elements g^,g^^G. Κ unitary representation U{g) is called cyclic, if there exists a vector he\) such that the smallest closed s u b s p a c e in i) which contains all vectors U{g)h,ge G , coincides with ί). Without loss of generality, we may suppose that || A || = 1. T h e vector h is called a cyclic vector for the representation U{g), We begin by considering cyclic representations of the group Φ. In other words, we consider continuous operator-valued functions υ{φ), whose values are unitary operators in a Hilbert space I), and i7(
h\,

where h is some fixed cychc vector of the representation [/(φ), and ( , )i denotes the scalar product in i). T h i s functional is positive-definite. Indeed, for any elements
η

η

η

But

(^(φ, -

φ,)Η, h), = {υ*{φ,)υ(φ,)Η,

h)^

= ((/(φ,)Α, ί/(φ,)Α), ,

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and therefore η

η

η

j=l

fc=l

η

i=l k=\ |2

>0

(II A 111 is the norm in i)). T h i s proves the positive definiteness of L{φ). It is further obvious that L ( 0 ) = (A, h)i = 1, and that in view of the continuity of the representation U{ψ),L{ψ) is continuous. Applying Bochner's theorem in nuclear spaces (cf. T h e o r e m 1 of Section 4 ) toL(
(14)

J u s t as in the spectral analysis of operators (cf. the appendix to Chapter I, Section 4 ) , one can prove that ^ can be realized as the space of functions / ( F ) , defined on Φ' and having square integrable moduli with respect to μ, in such a way that the operator which corresponds by this realization to the operator [/(φ) is the operator of multiplication by T h i s realization consists in associating with the vector Κ = %λ,υ{φ,)Η

(15)

k=\

in I) the function fm

= % λ,.^<^.^.)

(16)

on Φ'. It follows from ( 1 4 ) that this correspondence is isometric. Since A is a cychc vector, the vectors of the form ( 1 5 ) are everywhere dense in i), and therefore this correspondence can be extended to all vectors of i). Now the operator υ{φ) takes a vector of the form ( 1 5 ) into the vector k=\ to which corresponds the function

k=l

fc=l

5.4

Quasi-Invariant Measures

365

Consequently, on functions of the form (16) the operator corresponding to υ{φ) is the operator of multiphcation by e'^^^'^^K But it is easy to prove that these functions are everywhere dense in L^. Therefore the operator in corresponding to υ{φ) is the operator of multiplication by ^^<^·^>. T h u s we have proven the following theorem. T h e o r e m 5. L e t f/(
1{φ) ^ {υ{φ)Η, h) = j

άμ{Ρ),

(17)

T h e space I) can be realized as the space of functions f{F) on Φ' having square integrable moduh with respect to μ, in such a way that the operator corresponding by this realization to υ{φ) is the operator of multiplication by e^^^'^^K If one chooses a vector h^el) different from h (in general, will not be cychc), then to also there corresponds a positive-definite continuous functional Lι{φ), defined by

T h e functional Lι{φ)

is the Fourier transform of a positive measure

μι in Φ': Ιι{φ)

= j β^^^><Ρ^μι(Ε),

(18)

T h e measures μι and μ are connected by the relation άμ,(Ε)=\/{Ε)\^μ{Ε),

(19)

where f{F) is the function which corresponds by T h e o r e m 5 to the vector Ap Indeed, since to the operator υ{ψ) there corresponds the operator of multiphcation by ^^<^·^> in L^, and the correspondence between i) and is isometric, then 1,{φ)

= (ί/(φ)Α„Α,)ι =

|e^^^'^>|/(F)r^^K^).

Comparing this with (18), we conclude that (19) holds. It follows from (19) that if μ{Χ) = 0, then μι{Χ) = 0. If hi is also a cyclic vector, then the converse is true. T h u s the measures corresponding to diff^erent cyclic vectors in i) are equivalent to one another. Finally, we remark that given a normahzed measure μ in Φ', there

366

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exists a unitary representation υ{φ) of the representation such that

TOPOLOGICAL SPACES

Ch.

IV

of Φ and a vector h in the space

(υ(φ)Η, h) = j €'^^^^^μ{Κ). Indeed, denote by the space of aU functions f{F) on Φ ' having square integrable moduli with respect to /x, and associate with each φ e Φ the operator ί7(φ) i n L ^ which takes any function f{F) into the function e^^^-'^^ f{F). Obviously υ{φ) is the desired representation. Let us consider unitary representations of Φ which are not cyclic. In this case there is a finite or countable set {/x„} of measures in Φ ' such that i) is the direct orthogonal s u m of the spaces L^^ and to the operator υ{φ) there corresponds in each of the Lj, the operator of multiphcation by ^ ^ ' < ^ · 9 ^ >. F r o m this it follows that I) can be realized as a direct integral of Hilbert spaces 1) =

ί

@H{F)d^(F)

J Φ'

in such a way that to the operator υ{ψ) there corresponds the operator of multiplication by ^^^^·*^^ We shall not carry out the details of the corresponding arguments. We now turn to unitary representations of the group GQ. Recall that this group consists of elements.of the form (i>

Φι\ <^ι){ψ2> Φ2'> 0C2) =

{ψι

+

^2,

Φι +

Φ2\ e^'^^^^^^a^^^),

(20)

Consider the set Φ ^ in GQ consisting of aU elements of the form (
then this set of elements forms a subgroup in GQ which is isomorphic to the group Φ . In the same way, the elements of the form (0, φ\ 1) form a subgroup in GQ which also is isomorphic to Φ . Finally, the elements of the form (0, 0; a ) form a subgroup A in GQ which is iso­ morphic to the multiplicative group Τ of complex numbers of modulus 1. Let U{g) be a unitary representation of the group GQ. Restricting i7(^) to the subgroup Φ^, we obtain a unitary representation ί7(φ) of the group Φ . In the same way, the restriction of U{g) to yields another unitary representation ν{φ) of Φ (we denote by ν{φ) the operator U{g)

5.4

Quasi-Invariant Measures

367

corresponding to g of the form (0, i/r; 1)). Finally, the restriction of i7(^) to the subgroup ^4 is a unitary representation W{oi) of the group Γ . Since any element g = {φ, Φ\ oc) in GQ can be v^ritten as a product ( 9 > , 0 ; a ) = (9>,O;l)(O,0;l)(O,O;c.)

of elements of the subgroups Φι, and A, then the operator corresponding to g can be written in the form U(g) =

(21)

U{g)

υ{φ)ν{φ)ΐν{<χ).

Therefore to define i7(^) it suffices to specify the representations υ{φ), ν{φ), and W{oc). For simphcity we restrict ourselves to the case where the representa­ tion W(OL) of Τ has the form lV{(x) = a , and the representation υ{φ) is cychc (the general case can easily be reduced to this case). We show that in this case a complete description of all representations of GQ reduces to the description of ah pairs ( ί / ( φ ) , ν{φ)) of representations of the group Φ, satisfying the commutation relations ν{φ)υ{φ) = 6^^'Ρ^^^υ{φ)ν{φ).

(22)

Indeed, let U(g) be a unitary representation of GQ. It follows from (20) that (0, Φ; 1)(φ, 0; 1) =

( φ , 0; 1)(0, φ; 1)(0, 0; .^i^.v-)).

(23)

Since we have assumed that W{(x) = a , the proof of (22) follows directly from (23). Conversely, if the unitary representations υ{φ) and ν{φ) satisfy the commutation relations (22), then, taking

U{g) = ocU{φ)V{φ) for g = ( φ , 0 ; α ) , we obtain a unitary representation of GQ, Indeed, if ^ 1 = (9i> 0 1 ; OIL) and g^ = {φ^, Φ2\ « 2 ) , then ^{gxmg2)

= OLΙOC,U{ψι)V{φι)U{ψ,)V{φ,) =^.^<^.^ι)α,α2^(φι)ί/(9>2)ϊ^('Αι)Ι^(Ά2)

= β^^^^^^ύαια^υ^ψι + φ2)ν{φι + φ^) = Uigig^). We now turn to the description of all pairs {υ{ψ), ν{φ)) of representa­ tions of the group Φ which satisfy the commutation relations (22).

368

MEASURES

IN

LINEAR

TOPOLOGICAL

SPACES

Ch.

IV

Since we are considering only the case in which υ{φ) is cychc, by T h e o r e m 5 there exists a normahzed measure μ in Φ' such that (υ(φ)Η, h) = j e'^^^^^ αμ{Ρ) for all φΕΦ (A is a fixed cyclic vector of the representation υ{φ)). T h e space I) of the representation can be realized as the space Lf, of functions f{F) on Φ' having square integrable moduli with respect to μ, and to the operator ί/(φ) there corresponds in the operator of multiphcation by e^^^'^^K We now prove that for this representation ί7(φ) the operators ν{φ) are given (in the space L^) by ν{φ)/{Ε) = α^{Ρ)/{Ε + Γ^),

(24)

where ay,{E) = I^(iA)/o(^)>/o(^) — 1> is the hnear functional on Φ defined by (E^, φ) = (φ, φ). Indeed, since the operator inL^ correspond­ ing to υ{φ) is that of multiphcation by any function of the form fin

= X λ,/<^'-.>.

(25)

can be written in the form f{F) = X

λ,[/(φ,)/ο(ί').

k=l

Consequently, in view of the commutation relations we have

V{Φ)m

= X

Χ,ν(φ)υ(φ,)/,{Ρ)

k=l

= X λ , exp[/(
-= X λ ,

exp[/(9„

φ)]υ(φ,)ν{φ)/,{Ε) Φ)] exp[z(F, 9 , ) ] ^ , ( F ) .

Since exp [ί(φ^, φ)] = exp [i(F^„ φ;^)], this relation can be

written in

the form K ( 0 ) / ( F ) = a^{F) X λ , exp[i(F + F , , ^ . ) ] = s ( F ) / ( F + F , ) .

5.4

Quasi-Invariant Measures

369

As the functions of the form (25) are everywhere dense in L^, the relation F(^)/(F) = a,(F)/(F + F , )

(26)

holds for all functions f{F) in Ll. T h e functions ay{F) satisfy the functional equation + F^^Y

a,^,,lF)^a^lF)a^lF

(27)

Indeed, in view of (26) = ηΦ^ + ΦΜΡ)

=

ν{Φ^)ν(Φ,)ΜΡ)

= V{φ,)a.niP) = %^iPK,{P

+ Pn)•

We have found the realizations of υ{φ) and ν(φ) in L^. L e t us now show that the measure /x, corresponding to the representation υ{φ), is quasi-invariant (in Φ'). T o do this, we note that from the unitarity of ν{φ) it follows that

(/,/)i = m)f, for every

feLl.

vm\

T h i s can be written as

' | / ( F ) | 2 d^{F) =

j

|/(F +

F.,,)r^|

a,(F)|2

d^{F).

Replacing the variable F by F + F ^ in the left side, we obtain ' | / ( F + FXάμ{Ρ

+ F , ) = J | / ( F + F^l

Since (28) holds for all functions

f{F)eLl,

a^{F)\^άμ{Ρ),

(28)

then

άμ^{Ρ) ^ άμ{Ρ + F,,) = I a,(F)|2 άμ{Ρ).

(29)

T h u s , under translation by the vector Fy, corresponding to the element i/f 6 Φ, the measure μ is taken into the measure μ^ defined by

/x,(X)= ί

\α^{Ρ)\^άμ{Ρ).

Obviously μ^^Χ) = 0 if μ{Χ) = 0. But this means that μ is quasiinvariant. We have thus proven the following theorem.

370

MEASURES IN

LINEAR TOPOLOGICAL SPACES

Ch.

IV

T h e o r e m 6. S u p p o s e that U{g) is a unitary representation of the group Go, in a Hilbert space / / , which induces a cyclic (unitary) re­ presentation of the subgroup of GQ consisting of all elements of the form ( φ , 0 ; 1 ) , and the representation W{(x) = α of the subgroup of GQ consisting of all elements of the form ( 0 , 0 ; cx). T h e n there exists a quasi-invariant measure (in the sense of the definition on p. 3 5 5 ) μ in Φ' such that (^/(φ)Α,

h) =

j

β'^^^^^μ(Ρ),

where ψ is an element of the subgroup Φ^, and A is a cyclic vector of the representation υ{φ). T h e Hilbert space Η can be realized as the space of functions on Φ' which have square integrable moduli with respect to μ, in such a way that υ{φ) is the operator of multiphcation by e^^^'^^^ and ν{φ) is given by ν{φ)/(Ρ)

= α^(Ρ)/{Ρ + Ρ^),

where the ay,{F) are functions on Φ' satisfying the functional equation

andFy, is the element of Φ' such that (F^, φ) = ( φ , φ). Under translation by the vector Fy, e Φ', the measure μ transforms according to μ^{Χ)=

ί

\α^{Ρ)\^μ{Ρ),