POSITIVE CONTENTS AND MEASURES

CHAPTER I

POSITIVE CONTENTS AND MEASURES

The raison d'être of content and measure theory is the desire to measure the content (say, of volume, area, or mass) of a great variety of subsets of a given basic set Ω, and to "integrate" a great variety of, say, real functions on Ω. This desire is channeled by a very simple idea: It seems to be obvious how to measure the content of certain comparatively simple subsets of Ω (such as rectangles if Ω = U2 and we want to measure areas), or how to integrate certain comparatively elementary functions (such as step functions, which are constant on "simple" sets); the problem is how to extend this measurement resp. integration to more complicated sets resp. functions. Thus we are to define real functions ^ 0 on a certain domain consisting of subsets of Ω which we consider as "simple," and this leads to the notion of a (positive) content on a ring of subsets of Ω. Similarly, we are led to consider functionals on a certain set of real functions on Ω that we consider as "elementary" and this leads to the notion of a (positive) measure on an elementary domain of functions on Ω. The purpose of this chapter is a careful definition and discussion of these two notions and their simplest properties. For each of them we offer an extension procedure which we name after Eudoxos (ca. 408-355 B.C.) resp. Riemann (1826-1866) (Jordan (1838-1922) could compete as well). The discussion will show why these extensions are too simple in order to satisfy certain "higher" desires, and thus will motivate the more subtle extension procedures that are to be presented in Chapters II and III. 1. CONTENTS ON SET RINGS

1.1. Definition. Let Ω = {ω,rç,...} be an arbitrary set # 0 . A system (i.e., a set) 0t of subsets of Ω is called a (set) ring on (or in (the basic set)) Ω if: 26

1.

1.1.1.

CONTENTS ON SET RINGS

27

0t is stable under finite unions, i.e., => fi! υ · · · υ £ Η € #

Ei9...9Ene&

(Η = 0,1,...).

According to the usual convention that a union of 0, i.e., no sets, also called an empty union, is 0 ? our requirement implies 0 e 0t. In particular, ât is nonempty. 1.1.2.

0t is stable under differences, i.e. =>

E,Fe@

E\Fe0t.

A ring containing the basic set Ω is also called a (set) field. / / 0t is a ring in Ω, then the couple (Ω, 0ί) is also called a measurable prespace. 1.2. Example. The system ^(Ω) of all subsets of a set Ω Φ 0 is a set field in Ω. The system of all finite subsets of Ω Φ 0 is a ring in Ω, and a field iff Ω itself is finite. The system of all those subsets of Ω that are either finite or have a finite complement is a field in Ω. 1.3. Simple observations on set rings and fields. 1.3.1. Every ring in an Ω φ 0 is stable under finite intersections: Let Fl9 ..., Fne 3tt, put F = F 1 u - u F „ , then F e f and F1 n -- n Fn =

F\[(F\Fl)^-u(F\Fn)]e^.

1.3.2. Every at most countable union of sets from a ring 0t can be written as an increasing union of sets from 0ί\ If Fl9 F2, ...e0t, F = F 1 u F 2 u · · · , we put £„ = F 1 u - u F „ (n = 1, 2, ...) and get 0t^Ex ç £ 2 ç ..., EX u E2 u ··· = E. 1.3.3. Firsi entrance decomposition: This is a simple device which transforms every countable union of members ( = elements) of a ring 0t in Ω into a countable disjoint union of smaller members of 0t. Let Fl9 F2,... e 0t and put F 0 = 0 , £„ = *Ί u ··· u Fn, F'n = Fn\En.l (n = 1, 2, ...). Then clearly E0, El9...e£ and F\, F2, ...e@, F\^Fi9 F2^F2, ..., Fj n F'k= 0 (j Φ k) and (recall that + means disjoint union) En = F\ + - + F'n F

1

uF

2

(a = 1 , 2 , . . . )

u - = F 1 + F'2 + · · · .

The proof (by induction) is left to the reader. 1.3.4. Every field 3F in a set Ω Φ 0 complements : Fe&

=>

is stable under the passage to

[F = 0\F

G

J*\

28

I.

POSITIVE CONTENTS AND MEASURES

1.4. Example. form

Let Ω = R be the real axis. A half-open interval of the

[A

[2''

k+ l 2r

=

(XX G

with integers r, k and r ^ 0 is called a (half-open) dyadic interval of order r in R. 1.4.1. For every r = 0, 1, ..., the system Q)r of all finite disjoint unions of dyadic intervals of order r is a ring in R. 1.4.2. ^o ^ 0 1 ^ ' ' ' > and 0 = ^ 0 u 2l u · · · is the system of all finite unions of dyadic intervals of any order in R. It is a ring in R. We call (R, 3>) the dyadic measurable prespace on R. The proof for 1 and 2 (Hint: [k/T, (k + l)/2 r [ = [2k/2r+ \ (2* + l)/2 r + '[ + [(2k + l)/2 r + ', (2k + 2)/2 r+ »[)

is left to the reader. 1.5. Exercises. Let Ω Φ 0 be a set. Recall that a system (= set) ξ of nonempty subsets of Ω is called a partition of Ω if it consists of pairwise disjoint sets with union Ω. The elements (= members) of ξ are then also called the atoms of the partition ξ. A partition η is said to be finer than the partition ξ of Ω if every atom of η is contained in some atom of ξ. In this case we write ξ ^ η or η ^ ξ. Prove: 1.5.1. If ξ, η are two partitions of Ω such that ξ ^ ^y, then there is a unique partition τ of fy and a natural bijection between ξ and τ such that every atom of ξ is the union of those atoms of η that are elements of the corresponding atom of τ. η is called finite over ξ if every atom of τ is finite, i.e., if every atom of ξ is a finite union of atoms of η. 1.5.2. If ξ, η are any two partitions of Ω, then there is a unique partition p, also denoted p = ξνη and called the common refinement of ξ and η, of Ω such that ξ ^ p, rç ^ p, and p ^ σ for any partition σ of Ω such that ζ ^ Ö",rç^ σ. The atoms of p are exactly those A n B with A G ξ, Βεη that are nonempty. 1.5.3. Let Ω = R. The half-open intervals of order r form a partition <5r of R (r = 0, 1, ...) such that δ 0 = ^ι = "· a n d <5r+i is finite over (5r(r = 0, 1,...). 1.5.4. Let Ω # 0 be arbitrary again. If ξ is a partition of a nonempty subset Ω0 of Ω, then the system of all finite unions of atoms of ξ is a ring @(ξ) in Ω. If 0t' is a ring in ξ, then ^ = {[jAeEA\E e 3t'} is a ring in Ω. If Ω0 = Ω and $' is a field, then so is 01.

1.

1.5.5.

CONTENTS ON SET RINGS

29

Let Ω, Ω' and/: Ω -► Ω' a mapping. Then the sets of the form / - V ) = {ω|ω G Ω, f(œ) = ω')

(ω' ef(Q))

form a partition of Ω, the partition derived from /. If at' is a ring in Ω', then / " \0t') = {f~ l(F) = {œ\f(œ) e F'}\F e X} is a ring in Ω. If
Let n ^ 1 be an integer. For any x = (xu ...,x„), y = (yl9 . . . j J e R "

define x^y

iff

xi ^ ^ i , . . . , x n ^ ^ n ,

x
iff

Xi < y i , . . . , x „ < ^ .

For any a = (au ..., an), b = (bt, ..., bn) e Un, define [a, b[ = {x\x e U\ a ^ x < b} and call this the (possibly empty) half open interval (in Un) from a to fc. It can clearly be written as a cartesian product [ai9 b^ x · ·· x [απ, fe„[ of half-open intervals in R, and it is nonempty iff a < b. It is called dyadic of order r if [a1? /^[, ..., [an, bn[ are dyadic intervals of order r in U. Let <5; be the system of all half-open dyadic intervals of order r in Un. Prove that δη0 ^ δ*[ ^ · · · and δ% ί is finite over 6nr (r = 0, 1, ...). Put Dnr = R(ônr) (r = 0, 1, ...). From 1.5.7 we have @n0 <= &\ ç · · · and ^ n = ^S u ®\ u · · · is a ring in (Rn. In particular, ^ 1 is the 2 of Example 1.4. We call ((R", ^ n ) the dyadic (elementary, geometric) measurable prespace in Un. 1.5.9. Let Δ be a system of rings in Ω. Assume that Δ is increasingly filtered, i.e., that for any @t,
30

I.

POSITIVE CONTENTS AND MEASURES

1.6. Example. Let A Φ 0 be a finite set. We shall call it the alphabet in the present context. The set A x A x · · · = {ω = ω0 ωγ · · · | ω 0 , ω ΐ 5 . . . e A) of all one-sided infinite sequences (we deviate here from the usual way of writing sequences in some way, omitting some brackets and commas, for the sake of simplicity) of symbols from A is called the (one-sided) shift space or Bernoulli space over A. Clearly it has only one element if A has, and a continuum of elements if A has more than one element. For every r = 0, 1, ..., the sets of the form [k0,...,kr]

= {ω = ω0ωΐ -'\ω0

= k0, ωγ = ku ...,œr

= kr]

are called the (special) cylinders of order r in Ω. They form a finite partition yr of Ω (with | 4 | r + 1 atoms, where \A\ denotes the power of A, as usual). From [k0,...,kr]

= l)[k0,.:9kr,k] keA

(r = 0, 1,...)

we see that yo = 7i = ' " a n d TV+i is finite over yr (r = 0, 1, ...). If # r denotes the system of all finite unions of special cylinders of order r, then <€r is a field in Ω, # 0 ç # ΐ 5 ..., and # = ^ 0 u *Ί ' * * is also a field in Ω. It consists of all finite unions of special cylinders of any order (1.5.7). (A x A x ···, <$) is also called the Bernoulli or shift measurable prespace for the alphabet A. 1.7. Definition. Let (Ω, ^ ) be a measurable prespace. 1.7.1. A real function m on 0t is called additive if w ( F 1 + - - + Fll) = m(F 1 ) + - - + fii(Fll) holds for any pairwise disjoint Fu ..., F„ e 0t. 1.7.2. Λη additive real function m on St is called a (positive) content if it is nonnegative. The triple (Ω, ^?, m) is then called a content prespace (over Ω). 1.7.3.

If m is a positive content on the ring 01, then \m\ = sup m(F) F €0t

is called its total mass or total variation. 1.7.4. A content m on a ring 0t is called bounded if \m\ < oo, and normalized if\\m\\ = l.If\\m\\ = oo, m is called unbounded. 1.7.5. A content prespace (Ω, 0t, m) is called a probability prespace if Qet% (i.e., 0t is afield) and m(Q) = 1. In this case m is also called a probability (content).

1.

CONTENTS ON SET RINGS

31

1.7.6. Let (Ω, 0t, m), (Ω, 01', m') be two content prespaces with the same basic set Ω φ 0. If & c $' and m coincides with m on $, then (Ω, 0t', m') is called an extension of(Q, 0t, m) (and m! an extension of m from 0t to $', of course). It is called a proper extension if M Φ 0ί'. 1.7.7. A function m: l - > R + = R u {oo} or a function m: #-► R_ = M u { — oo} is called additive if (F lf ..., F . e « , m(Fx) + · - + m(Fn) = m(Fx) + ··· + m(Fn)

FjnFk=0(j*k)).

1.8. Example (point mass distribution). Let 0t be a ring in Ω # 0 . Choose any ω1? ..., ωη e Ω, α = (al9 ..., a„) G R". Then

m(F) = Σ a*

( F e ^)

defines an additive real function m on 01. If a ^ 0, then m is a content with total mass o^ + · · · + a„. Assume that ^ separates the points ω 1 ? ..., ωη, i.e., that for any 1 ^ j , kr^n, j φ k, there is either an E G 0t with coj, e E $ œk or an F e 01 with œke F $ω}. Then different a e R " lead to different m, and m is a content iff α ^ 0. It is easy to generalize this to countably many points ω ΐ5 ω 2 , . . . e Q and reals α1? α 2 , ... with |αχ I + |α 2 | H- ··* < oo. It is equally easy to generalize this to an arbitrary family (œt)ieI of points œteQ and an arbitrary family (a,) l6/ of reals, provided \{i\œte E}\ < co (E e âl). 1.9. Simple observations about additive set functions: 1.9.1. Since 0 G 0t for every ring 0t, additivity of m:âl->U implies It also implies

m(0)

= m(0 + 0 ) = m(0) + m(0) = 0.

m(E u F) + m(£ n F) = m(F + (F\F)) -f m(F n F)) = m(F) + m(F\E) + m(F n F) = m(F) + m(F). 1.9.2.

Every additive function m on a ring 0t is isotone, i.e., E,FeR,E^F

=> m(F) ^ w(F).

In fact, m(F) = m(E + (F\E)) = m(E) + m(F\£) ^ m(F). 1.9.3.

Every content m on a ring 0t is subtractive, i.e., m(F\F) = m(E) - m(F)

(E, F e @, F => F).

In fact, m(E) = m(F + (E\F)) = m(F) + m(E\F). 1.9.4. Every content m on a ring 0t is subadditive, i.e., m ^ u • • u F n ) ^ m ( F 1 ) + "- + m(Fn) ( F l f . . . , Fn G Λ).

32

I.

POSITIVE CONTENTS AND MEASURES

In fact, if we construct, according to the first entrance decomposition 1.3.3, ..., F'n^Fn of 0t such that pairwise disjoint members F\^FU F\ + · · · + F'n = Fx u · · · u Fn, we see m{F, u ··· u Fn) = m(F\ + ··· + F'n) = m(F;) + ··■ + m(Fn) £m(Fl) + - + m(Fn). 1.9.5. Any finite linear combination of additive real functions on a ring 9t in some Ω Φ 0 is additive again. Any finite linear combination with nonnegative coefficients of contents on $ is a content on 0t again, and UttiW! + --- + anmn|| =a 1 ||m 1 || + --- + a„||mj in this case. If M is a field, the probabilities on ^2 form a convex set, i.e., a set that is stable under finite linear combinations with nonnegative coefficients summing to 1. 1.9.6. Observe that 1.7.7 makes sense since the possibility of a pathological addition oo + ( — oo) is excluded. If we modify Example 1.8 such that 0Lk = oo for some /c, we get a simple example of additivity with value oo admitted, m = oo would be another example. The only cases of real interest are those for which m(0) = 0. 1.10. Example. 1.4) put

Let Ω = R. For every r = 0, 1, ..., FeQ)r (Example mr(F) = c/2'

if F is a disjoint union of c dyadic intervals of order r (clearly F uniquely determines c). Then m is an unbounded content on 3)r. Since every dyadic interval of order r can be represented as a disjoint union of two dyadic intervals of order r + 1, we see mr(F) = jr = ^

= mr+l(F)

(Fe3r\

and this implies that m r+1 is an extension of mr from Q)r to @r+i (r = 0, 1, ...). Consequently, there is a unique real function m on 2 = @0 u ® l u ' * * s u c h that the restriction of m to @r is mr (r = 0, 1, ...). m is an unbounded content on 3). It is called the elementary, geometric, Riemann, Jordan, or Lebesgue content on ®, and (R, Q), m) is called the dyadic elementary etc. content prespace over R. 1.11.

Exercises

1.11.1. Let ξ be a finite partition of a nonempty subset of Ω Φ 0 , and a a real function on ξ. For every F G ^(£) (Exercise 1.5.4) let m(F) = E ^ S F a(^)· Prove that m is an additive function on &{ξ) and that it is a content iff α ^ 0, its total mass then being Υ^ €ξ ct(A).

1.

CONTENTS ON SET RINGS

33

1.11.2. Use Exercises 1.5.7, 8 and carry Example 1.10 over to the dyadic elementary measurable prespace (Un, Dn) in such a fashion that m([a, b)) = \/2nr for any dyadic half-open interval [a, b[ of order r in W (n = 1, 2, ...). The resulting content prespace ((Rn, D", m) is called the dyadic elementary etc. content prespace on Un. 1.11.3. Let Δ be an increasingly filtered system of rings in Ω Φ 0 . For every R e Δ, let m^ be a content on 0t. Assume that for 0t, if e Δ, 0t ç if the restriction of m#> to 0t coincides with m#. Show the unique existence of a content m on the ring \)^^^^ (Exercise 1.5.9) such that for every ^ e A , the restriction of m to 0t is m#. 1.11.4. Use 3 and carry Exercise 2 over to the situation envisaged in Exercise 1.5.10,providingmi\aubx[ x ··· x [an9bH[) = (bl - ax) ··· (bn - an) for every half-open interval [au bx[ x ··· x [an, bn[ £ Un. The resulting content prespace is called the elementary etc. content prespace on Rn. 1.12.

Example.

LetA^

0 be afiniteset. A real function p on A such that p(fc) ^ 0

(keA)

is also called a probability vector (over, or indexed by /I). Let Ω = A x ,4 x · · · be the (one-sided) shift space over A and ^ 0 ^ *Ί ^ ' " * ^ * as in Example 1.6. For every r = 0, 1, ..., there is a unique probability content mr on (€r such that m r ([/c 0 ,..., kr]) = p(k0) · · · p(kr)

(k0,...,kre

A).

(Use Exercise 1.11.1) or direct reasoning.) From mr+ iflTco,..., kr]) = £ mr+l([k0,..., fce/l

kr, fc])

= IP(*I)-P(MP(*)

= ρ(^)···ρ(ΜΣ#) fce>l

=

p{k,)'-p{K)

= m r ([/c 0 ,...,/c r ]) we infer that mr+l is an extension of mr from ^ r to # Γ +ι· Thus (Exercise 1.11.3) there is a unique probability content ΐίΐοη^ = ^ 0 υ ^ υ · · · such that the restriction of m to ^r is mr (r = 0, 1, ...), m is called the Bernoulli content for (or associated with the probability vector) p and (Ω, #, m) the corresponding ("one-sided") Bernoulli probability prespace.

34

I.

1.13. 1.13.1. 1.12.

POSITIVE CONTENTS AND MEASURES

Exercises Carry out an obvious "two-sided" analogue of Examples 1.6 and

1.13.2. Carry out an analogue of Example 1.12 where the probability vector p may "vary from component to component." Carry this over to the situation of Exercise 1. 1.14. Example. A x A such that

Let A, Ω, p as in Example 1.12. A real function P on (a)

P ^ 0, i.e., P(j, k)^0

(b) Σ η / . * ) = 1 keA

(/,

keA)

UeA).

is also called a stochastic matrix (over, or indexed by A). For every r = 1, 2, ..., there is a unique probability content mr on (€T such that m r ([/c 0 ,.... kr]) = p{k0)P(k0, kx) ··· P{kr.l9 kr)

(/c 0 ,...,

kreA).

From Mr + 1 ([ko , . . -, M ) =

Σ ™r + 1 (|>0 , · · ·, fcr , *] )

= p(k0)P(k0, *,) ··· Ρ(ΛΓ_!, /cr) Σ i 5 ^ , *) = p(k0)P(k0,ki)'~P(kr-u

K) = mr([k09...,

M)

(

we infer that m r + 1 is an extension of mr from €r to ^ Γ + ι · The unique content w i o n ^ = C 0 u ( ^ 1 u · · · whose restriction to ^ r is mr is called the Markov content for (or associated with) the initial distribution p and the transition matrix P, and (A x A x · · ·, ^ , m) is then also called the associated Markov probability prespace. 1.15. Exercise. Carry out an analogue of Example 1.14 where the transition matrix P "may vary from transition to transition."

2. ^-CONTENTS

In many cases a content has a property that is sharper than additivity: σ-additivity. This will turn out to be the basis of the sophisticated extension procedures to be discussed in Chapters II and III. In this section we set up its definition and discuss its occurrence and most direct implications.

2.

σ-CONTENTS

35

2.1. Definition 2.1.1. Let (Ω, 0t, m) be a content prespace. The positive content m is called σ-additive or a σ-content if m(Fl + F2 + (F1,F2,...e«,

) = m(Fl) + m(F2) + · · ·

F,.nFk = 0(/#/c),

F1+F2

+

-e&)

(recall that + also denotes disjoint union). In this case (Ω, 0t, m) is also called a σ-content prespace. / / a content m is a probability, we call it also a ^-probability and speak of the ^-probability prespace (Ω, 0t, m). 2.1.2. A σ-content prespace (Ω, 0t, m) is called full if Fx, F 2 , ...e0t, m(Fl) + m(F2) + · · · < oo implies Fx u F2 u · · · G $. 2.1.3. Let £f be a nonempty system of subsets of ίϊφ 0. A function m : y - » R + = R u {oo} or m: M_ = R_ u {- oo} is ca/ted σ-additive i/ /or any Fu F2,... G ^ swc/z ί/ιαί F,· n F k = 0 0 V fc) α ^ F x -h F 2 -h · · · G «^, ί/ie series m(Fl) + m(F 2 ) + · · · (whose members may attain one of the values ± oo), converges to the (possibly infinite) value m(Fl + F 2 + · ··). An easy criterion for σ-additivity is given in 2.2. Proposition. Let (Ω, ^2, m) be a content prespace. Then m is σ-additive iff m is " σ-continuous at 0 " , i.e., iff (1)

m(Ffc)-0

( i 3 £ i = i F 2 =>...,

···= 0).

ElnE2n

Proo/. I. Let m be σ-additive and El9 F 2 , . . . G ^ , £ 1 ^ F 2 3 · · , F J r\.£ 2 n · · = 0 . Then Fj = (E^E^ + (F 2 \F 3 ) -f ··· represents Fi G ^? as a disjoint countable union of members of 0t, hence m(E,) = m(El\E2) -l· m(E2\E3) ^ Now Ffc = £ k + 1 4- (Ek\Ek+1) we get

-".

implies m(Ffc) = m(Ek+1) + m(Ek\Ek+1),

and

m ^ ) = H f i i ) - m(F 2 )] + [m(£ 2 ) - w(£ 3 )] + " · ' =

lim[m(E1)-m(En)] n-*ao

= m(El) — lim m(En); n-* oo

and hence m(Ek) -► 0, the convergence being a decreasing one, by the way. II. Let m satisfy (1) and let F x , F 2 , . . . G ^ be pairwise disjoint with Fx + F2 + -·- e 0t. Then we put Ek = Fk-\- Fk+l + ··· and see that

36

I.

POSITIVE CONTENTS AND MEASURES

and Ei n E2 n · · · = 0 . Thus by (1) we have m(Ek) -► 0 and therefore m(F1 + F2 + ···) = m(F1 + ··· + Ffc_x + Ffc)

i.e., m is σ-additive. 2.3. 2.3.1.

= m(F1) + "- + m(FJk-1) + m(£fc) ->ni(F1) + m(F2) + - ,

Exercises Prove that the contents defined in Example 1.8 are σ-additive.

2.3.2. Let (Ω, ^2, m) be a content prespace. Prove that m is σ-additive iff m is "σ-continuous from above" (i.e., F,F1,F2,...e^,F1:2F2^--,F1nF2n-=F

=>

m(Fn)^m(F)

=>

m(Fn)^m(F)).

iff m is "σ-continuous from below" (i.e., F,F1,F2,...,ef,F1çF2ç-,F1uF2u- = F

2.3.3. Let ^ be a ring in Ω # 0 and m: ^ - > K + additive. Prove that m is σ-additive iff it is "σ-continuous from below" (which is defined as in 2 but with values oo allowed), and that σ-additivity implies "restricted cr-continuity from above," i.e., 0t s Fj 2 F 2 2 · · ·, Fx n F 2 n · · · = F G ^2, lim m(F„) < oo => lim m(F„) = m(F). Prove that the latter property is equivalent to "restricted σ-continuity at 0 , " i.e., its own special case with F = 0 . 2.3.4. Let J be a σ-field in Ω Φ 0 and m: @ ->R + σ-additive. Let ^oo = |/r| j7 G ^ m(F) < oo} and let m also denote the restriction of m to ^ 0 0 . Prove that (Ω, ^ 0 0 , m) is a full σ-content prespace. 2.3.5. Let Ω = r^J = {1, 2, ...}, 0t = ^(Ω), and m(F) = oo if |F| = oo, m(F) = £ „ e F 1/2" if |F| < oo. Prove that m is additive but not σ-additive. 2.3.6. Let Ω, 0t be as in 5. Prove that m: F^ \F\ is σ-additive, but not "σ-continuous at 0 . " (Hint: Consider F„ = {w, n + 1,...}.) 2.3.7. Let Ω, ^ be as in 5. Put a„ = l/2 n/2 for n e ß , n even, a„ = 1 for n e Ω, n odd. Prove that m(F) = ^ 6 f a „ (F e 3$) attains every value in R+ and is σ-additive on 0t. 2.4. Example. Let (R, ®, m) be the dyadic geometric content prespace on R (Example 1.10). Then m is σ-additive. We prove this by a compactness argument, employing Proposition 2.2. Let Clearly the m(Ek) form a decreasing sequence of reals. Assume their limit is some a > 0. For every k = 1, 2 , . . . , we construct some E'k ç Ek such that: (a) m(E'k)>m(Ek)-x/2k+1; (b) the closure Ek of Ek is contained in Ek.

2.

σ-CONTENTS

37

This is an easy thing if Ek is a dyadic interval [k/2", (k + l)/2"[ of some order n: Choose r>n and observe that E„ =

k2r-" + 2r -n 2'

kl'-

~Y

k2'-n + 1 + ··· + 2r

kl'2'

k2r-n

+

γ-η

2

_

{

^r-n

r

γ

+

r

2

r

represents Ek as a disjoint union of 2 " dyadic intervals of order r. Leaving off the rightmost of them, we define 2'-"-2

£ί=

Σ

k2r-n+j fc2r-"-h7-l· 1

j=0

and get E'k e <2)r

E'k =

A

k+ 1

ç£k,

m(£;) = m(£k) -

1

If r is large enough, we get l/2r < a/2fc+1, and we are through for this special case. If Ek is, say, in Q)n, i.e., a disjoint union of finitely many (say c) dyadic intervals of order n, we apply the above "curtailing" procedure to each of the latter and get E'k e 3)r, E'k ç Ek, and m(Ek) = m(Ek) — (c/2r) which is >m(Ek) — a/2fc+ * for r sufficiently large. Put now Ek = E\ n ··· n £^. Then ® 3 Ê ; ' Ç £;, hence 2%' ç £fc. We have £,\£*s(£i\£'i)^---^(£*\£i). hence m(£»VEÎ) â "ΐ((ΕΛ£Ί) u ··· u (£ t \£i)) im(El\E'1) + --- + m(E„\E'n) a Consequently

a a TT< 2* 2"

m(£0 = m(£fc) - m(£fc\£*) > a ~ 2 = 2 > °' which implies that E'[, El,... are nonempty. Thus £ï, Ë"2,... is a decreasing sequence of nonempty compact sets. A well-known topological result says that E\ n E'2 n '- Φ 0 (prove it as an exercise). The still larger set E1 n E2 n · · · is therefore also nonempty, in contradiction to our assumption. By Proposition 2.2 m is σ-additive.

38

I.

POSITIVE CONTENTS AND MEASURES

2.5. Example. Let (A x A x · · · , #, m) be the Bernoulli probability prespace for a given probability vector p over the finite set ΑΦ 0 (Example 1.12). Then m is σ-additive. For this, choose any <€ 3 Ex Ώ E2 3 · · · with lim*..^ m(Ek) = a > 0. If we can prove El n £ 2 n · · · Φ 0 , we are through by Proposition 2.2. We shall in fact prove it by a compactness argument which turns out to be simpler than that used in Example 2.4 because we can exploit the finiteness of A. Choose œ(k) = œ^}œ{k) · · · G Ek (k = 1, 2, ...) arbitrarily. For every t = 0, 1, ..., the component sequence œ\k) (k = 1, 2,...) runs in A, hence contains a finally constant subsequence. By an obvious diagonal procedure we may pass to a subsequence œ(kl\ ω(Ιί2\ ... of ω(1), ω<2), ... such that for each t = 0, 1, ..., the sequence a)\kn) (n = 1,2,...) is finally constant, say = œt. Put ω = ω0 ωί · · ·. We show ω G Ex n E2 n · · ·, i.e., ω G Ek (k = 1, 2, ...). For any /c, we represent Ek as a member of some ^ Γ , namely, as the disjoint union of cylinders of order r. Every œ{k\ ω (Λ+1) , ... belongs to some of these cylinders, thus every œ(kn) with n sufficiently large does so. If we choose n so large that ω(0*π) = ω 0 , ..., œ{kn) = œr, it follows that [ω 0 , ..., ωΓ] is one of the cylinders constituting Ek, and thus ω e Ek follows. This does it. The reader sees that we have not used m at all. Thus our proof applies to the effect that any content on # is σ-additive. The topologist sees, of course, what the reason is: All members of %> are compact subsets of the compact metric space Ω = A x A x · · ·, hence # 9 £ x 3 £ 2 — " > Ex c\ E2r\ ·- = 0 implies Ek = Ek+ ! = · · · = 0 for some k, by a well-known theorem from topology. 2.6.

Exercises

2.6.1. Carry over the result of Example 2.5 to the "two-sided infinite" situation envisaged in Exercise 1.13.1. 2.6.2. Carry over the result of Example 2.4 to Ω = Rn (Exercise 1.5.8, Exercise 1.11.2). 2.6.3. Carry over the result of 2 to the situation envisaged in Exercises 1.5.10 and 1.11.4.

3. EUDOXOS EXTENSION OF CONTENTS In many cases where a content prespace (Ω, ^, m) is given it seems desirable to "measure by m" some sets that are not yet in ^2, as we have said in the introduction to this chapter. Let, e.g., (R2, ^ 2 , m) be the dyadic geometric content prespace on 2 U (Exercise 1.11.2). The unit disk D = {(x, y)\x2 + y2 g 1} is not in 92, and even a set as simple as {(x, y)\0 ^ x, y < \) is not, to say nothing of

3.

EUDOXOS EXTENSION OF CONTENTS

39

a set like {(x, y)\x9 y rational}. The ancient problem of "measuring" the unit disk D (and hopefully to find m(D) = 3.14···) is an example of an extension problem: We have to find an extension (IR2, 0t, m) of (!R2, S)2, m) such that D e l Let us consider another example that stems from probability theory. Let A φ 0 be finite and p a probability vector over A. Let (A x A x · · ·, # , m) be the Bernoulli σ-content prespace for p (Examples 1.12, 2.5), specify any k e A and put E = {ω = ω0ωι · - \ œ t = k for at least one t = 0, 1, ...}. Then E is not in # if \A\ ^ 2 (exercise); but probabilists would, of course, like to ascribe a "probability" to it, that is, to find an extension (A x A x · · ·, 0t, m) of (A x A x ·-,<£, m) such that E e 01. We shall now present a rather simple extension procedure which goes back more or less to Eudoxos (ca. 408-455 B.C.) and Archimedes (287-212 B.C.) and can be described by the following: 3.1. Definition. Let (Ω, 0t, m) be a content prespace. A set F ç Q is called Eudoxos (or Jordan) measurable for (Ω, 0t, m) or, briefly, for m if for every ε > 0 there are F, G e 0t such that E^F^G,

m(G\E) < ε,

or, equivalently (obey the rule inf 0 = oo, sup 0 = — oo) — oo <

sup m(E) =

inf m(G) < oo. F Ç Ge&

0tsE^F

Let 0t{m) denote the system of all Eudoxos measurable sets (for (Ω, 0t, m)). 3.2. Theorem. Let (Ω, ^2, m) be a content space and ${m) the system of the corresponding Eudoxos measurable sets. Then: 3.2.1. 0t{m)^L0land m(F)=

sup m(E)

0tsE^F

(F e ®(m))

defines an extension (again denoted by m) of m from 0t to @{m). 3.2.2. (Ω, @(m), m) is a content prespace again. It is a σ-content prespace ijf(Q®,m)is. 3.2.3. Every nonnegative additive extension of m from 0t to ${m) coincides with the m given in 1, and ($(m)\m) = 3t{m). Proof. 1. If F e m, then clearly m(F) =

sup m(E) =

3tsE^F

inf m(G).

F Ç Ge£

2. We start by proving that ${m) is a ring. Let F, F' G &m\ we show that F u F' s @(m). Let ε > 0 be arbitrary and F, G, F', G e m be such that

40

I.

POSITIVE CONTENTS AND MEASURES

F ç F ç G , F ç F < = G ' and m{G\E) < ε/2 > m(G'\F). Clearly E u F, GuG'elJuFçFuFçGuG', m((G u G')\(E u £')) ^ m((G\E) u (G'\F)) ^ m(G\F) + πιψ'ψ') < ε/2 + e/2 = ε. The proof for F n F' e J?(m) goes similarly and is left as an exercise to the reader. Next we show that the extended (according to 1) m is additive again (its nonnegativity is obvious). Let F, F e 0t{m) be disjoint. Then i 9 £ ç f , f 9 £ ' ç F implies E n E' = 0. Thus we obtain m(F

+ F) =

sup

^ ^

m(H)

sup

m(E + F )

sup

[m(£) + m(F)]

j»3£ÇF,âf3£'£F'

= sup m(£) +

sup m(E')

= m(F) + m(F'). But we also have m(F + F ) =

inf

m(tf)

F+ F'Ç/fe^

^ S =

inf

m(G u G)

inf

[m(G) + m(G')]

F£Ge<»,F'ÇG'e<3

inf m(G) +

F^Ge0t

inf m(G')

F'^G'e£

= m(F) + m(F') and thus m(F + F ) = m(F) + m(F') follows. Let finally m be σ-additive on ^2; we prove that its extension m to 0t{m) is σ-additive there. For this we use the criterion given in Proposition 2.2. Let 0t{m) 3F1^F2^.'",FlnF2c\--' = 0. For a given ε > 0, choose E'u F 2 , ... e <22 such that Ek^Fk, m{E'k) > m(Fk) - e/2k (k = 1, 2, ...). Put Ffc = Fx n · · n Ffc (fc = 1, 2, ...). Clearly and thus

F;ç=Fku(Fk_aF;-i)u·-^^^) m(E'k) ^ m(Fk) + e/2k~x + · · · + ε/2 < m(Ek) + ε,

3.

EUDOXOS EXTENSION OF CONTENTS

41

hence m(Ek) > m(Fk) - 2ε (k = 1, 2, ...). But we have « 9 £ 1 2 £ 2 2 - , £ j ς F l 5 E2^F2, ..., hence El n E2 n "· = 0, which implies linifc^ m(£fc) = 0 by the σ-additivity of m and Proposition 2.2. We conclude lim^«, m(Fk) g 2ε. Since ε > 0 was arbitrary, l i m * ^ m(Fk) = 0 follows, and the σ-additivity of m follows from Proposition 2.2. The rest is obvious. 3 is an easy exercise. 3.3. Exercises 3.3.1. Let (IR2, ® 2 , m) be the dyadic geometric content prespace on IR2 (Exercise 1.11.2). Prove that every subset of (R2 whose boundary is a finite union of smooth curves (such as the unit disk {(x, y)|x 2 + y 2 ^ 1} is Eudoxos measurable for m. 3.3.2. Try to find weaker sufficient conditions for geometric Eudoxos measurability of subsets of U2 than that given in 1. 3.3.3. Carry over 1 (and possibly 2) over to arbitrary dimensions n = 1,2,.... 3.3.4. Let (A x A x · · · , C, m) be the Bernoulli σ-content prespace for a probability vector p over the finite set A Φ 0 (Examples 1.12, 2.5). Assume \A\ ^ 2 and p(j) > 0 (je A). Prove that for every fixed k e A, the set E = {ω = ω0ωι · -\œt = k for some t = 0, 1, ...} is Eudoxos measurable for m with m(E) = 1. (Hint: Start with the case \A\ = 2 and look at the complement of E.) 3.3.5. Consider (IR2, D 2 , m) as in 1 and prove that its Eudoxos extension (IR2, D2m), m) is translation and rotation invariant in the following sense: Two congruent subsets of IR2 (i.e., two subsets that coincide after suitable translation and rotation) are either both in Dfm) or both not in Dfm) and yield the same value for m in the first case. (Hint: Begin by considering intervals and prove translation invariance; observe that a rotation about the origin carries the unit disk into itself.) 3.3.6.

Carry 5 over to arbitrary dimensions n = 1, 2, ....

3.4. Remark. The above exercises show that Eudoxos extension settles our problems in many cases. It does not settle, e.g., the following problem: Let (IR, 2, m) be the dyadic geometric content. We would like to "measure" the set F = {x\0 ^ x < 1, x rational} (namely, with m(F) = 0 since it is a countable union of one-point sets which are clearly Eudoxos measurable with content value 0). But F is not Eudoxos measurable: Since F is dense in [0, 1[, any finite union of half-open dyadic intervals covering F must also cover [0, 1[ (exercise), but any half-open interval contained in F must be empty. Thus we find ourselves induced to search for an extension method that is more efficient than Eudoxos' device.

42

I.

POSITIVE CONTENTS AND MEASURES

4.
In this section we present a variety of types of systems of subsets of a given basic set Ω which seems to be more feasible for a sound formulation of an extension problem for σ-contents, and for an attempt to its solution than the concept of a ring or field. The leading point of view for the ideas staged in the next definition may already be guessed from Definition 2.1 of σ-additivity of a content: It is awkward to always have to check whether a certain countable union of sets from the ring 0i is again a member of 0t. This was also the main point in Remark 3.4. 4.1. Definition. A set ring 0t w Ω # 0 is called: 4.1.1. a (set)
=> F1KJ F2u

-e^;

4.1.2. a (set) σ-field (in Ω) if it is a σ-ring and afield (i.e., Ω e 0t)\ 4.1.3. a local σ-ring or a d-ring if for every Ω 0 e 01, the restriction { F | Q 0 2 F G ^ } of 0t to Ω 0 is a σ-field in Ω 0 , i.e., if 0t is stable under bounded countable unions, i.e., iffor every Ω 0 e 0t Q0^Fl,F2,...e@

=> F x u F 2 u

"-eat.

4.1.4. If$00 is a local σ-ring in Ω, then the couple (Ω, ^ 0 0 ) is called a local measurable space. / / 01° is a σ-ring in Ω, then (Ω, J· 0 ) is called a measurable space. //(Ω, &00, m) is a σ-content prespace such that (Ω, ^ 0 0 ) is a local measurable space, i.e., @00 is a local σ-ring, then (Ω, &00, m) is also called a σ-content space. If m is a σ-probability here (hence in particular $00 is a σ-field), (Ω, 0$00, m) is also called a σ-probability space. 4.2.

Remarks

4.2.1. Note that we speak of measurable spaces instead of prespaces as soon as stability under bounded countable unions is secured. Clearly a full σ-content prespace is automatically a (full) σ-content space. See Exercise 2.3.4. 4.2.2. σ is, as already in Definition 2.1, by tradition a symbol for the occurrence of countable unions. Similarly, δ is the traditional symbol for the occurrence of countable intersections. Our synonymity of "local σ-ring" with "<5-ring" is underlined by the following fact: A ring is a local σ-ring iff it is stable under countable intersections. In particular, every <5-ring is stable under countable intersections.

4.

σ-RINGS, LOCAL σ-RINGS, AND σ-FIELDS

In fact, if ^ is a local σ-ring in Ω φ 0 Ω0 = El9 Fk = Ω0\£* (k = 1, 2, ...); then

and Eu £ 2 , ...e0t,

43

put

Ê! n £ 2 n · · · = Ω 0 \(£! u F 2 u · · ·) e &. Likewise, if ^ is a ring th^t is stable under countable intersections, Ω 0 , Fu F 2 , . . . € « , Q O S F L F 2 , ..., put ΕΛ = Ω 0 \£ λ (fc = 1, 2, ...); then F x u F 2 u · · · = Ω 0 \(£ι n F 2 n · · ·) e 01. 4.2.3. Every σ-field is a field and a σ-ring. Every σ-ring is a local σ-ring. Every local σ-ring is a ring. 4.3.

Example.

Let Ω be an infinite set. Then:

4.3.1. The system of all (at most) countable subsets of Ω is a σ-ring in Ω, and a σ-field iff" Ω is countable. 4.3.2. The system of all subsets of Ω that are either (at most) countable or have an (at most) countable complement is a σ-field. 4.4. Example. Let Ω = U. The system of all (at most) countable conditionally compact subsets of M is a local σ-ring but not a σ-ring in U. Wherever in mathematics a certain type of object is defined by stability under a certain family of compositions, the idea of generation shines through. It always follows the same pattern, which we display in the next proposition for those special cases that we have under consideration right now. 4.5. Proposition. Ω Φ 0 . Then:

Let 9

be a (possibly empty) system of subsets of

4.5.1.1. The system of all local σ-rings in Ω which contain 9 is nonempty: it contains 0>(ίϊ) = {F\F ç Ω}. 4.5.1.2. The intersection of an arbitrary system of local σ-rings in Ω is again a local σ-ring in Ω. 4.5.1.3. In particular, the intersection of all local σ-rings in Ω containing 9 is a local σ-ring in Ω: the smallest local σ-ring (in Ω) containing 9. It is denoted by 0I°°(9) and called the local σ-ring generated by 9'. It equals £f if' Sf is already a local σ-ring. 4.5.2. Analogous statements hold for rings, σ-rings, and σ-fields. 4.5.2.1. 0t(9) denotes the ring generated by 9'. 4.5.2.2. &(9) denotes the field in Ω generated by 9. 4.5.2.3. @°(9) denotes the σ-ring generated by
44

I.

POSITIVE CONTENTS AND MEASURES

4.5.3. If Sf ç f g 0>(Ω), then: 4.5.3.1. @(¥)^@(3Γ). 00 4.5.3.2. ^ ( ^ ) s ^ 0 0 ( ^ ) = Λ00(Λ(.Γ)). 4.5.3.3. ^ ° ( y ) ç Λ°(^) = ^°(^ 00 (,Τ)) = a°(9t(P)). 4.5.3.4. The proof is an easy exercise and left to the reader (use, e.g., Remark 4.2.3.). 4.6.

Remarks

4.6.1. We emphasize once more that the generated system, no matter of what type, increases if the generating system !¥ increases: Sf £ £f' implies M(Sf) £ 9t{SP), etc. We generally say "countably generated" instead of "generated by a countable 9T 4.6.2. Whenever in mathematics we have a generation definition like 4.5, the next question is whether the elements of the generated item can be constructed in a more or less explicit manner from the generating system. In our case the question is natural whether, e.g., every member of 0${Sf) can be expressed by a finite chain of countable unions and intersections, and possibly some differences, through countably many members of Sf. We state here without proof that this is generally not the case. We shall deal at length with this problem in Chapter XIII. Here we offer a few special results in the form of the next proposition and the exercises following it. 4.7. Proposition. Let Ω # 0 . 4.7.1. Let 0tbe a ring in Ω. For every Ω0 e 01, let 01 η Ω0 denote the ring {F\0t 3 F ç Ω0} and 0ίΩο the σ-ring in Ω0 generated by it. Then ®00(®)=

(1)

U ^n„.

In particular, every E e Jf 00 (^) is contained in some F e0t. 4.7.2. Let @m be a local σ-ring in Ω. Then 4.7.2.1. ^°(^ 0 0 ) = { F 1 u F 2 u · · · \Fi,F2,...e^00} = {E1 + E2 + ---\El, E2,... eâS00, Ej nEk=0(j* w

00

4.7.2.2. E e @(@ ), Fe<% ^EnFe E^Ffor some Fe@00^Ee 3800. 4.7.3. Let @° be a σ-ring in Ω. Then:

k)}.

0S°°. In particular E e @(@00),

4.


4.7.3.1.

Λ(Λ°) = {F\Fe @° or Q\F e <%0}.

4.7.3.2.

E e J ^ ° ) , F e @° => E n F e a0.

4.7 .4. 4.7.4.1.

45

For every (Ω) we ftûtœ: &{
4.7.4.2. Λ(5^) ç ^ 0 0 ( y ) ç a\!f) ç Λ(5^). 4.7.5.1. Lei Σ be an increasing σ-filtered system of o-rings in Ω: 0$\, »\, . . . 6 Σ = > « Ϊ u i 5 u ··· <^@° for some # e l 77ΐέ?η U # 0 e i ^ ° w « σ-rfnöf. 4.7.5.2. Let
J(el

JieZ 00

Proof 1. Since Ω 0 ε & η Ω 0 , we have <#Ωο = # (Λ Ω ο ) Ç ^ 0 0 ( ^ ) (Ω0 G ^?) and thus ^ holds in (1). But the right-hand member in (1) is a local σ-ring containing 0l\ If F, Fl9 F2,... e ( J ß o e Ä ^ Ω ο , F l 5 F 2 , ... ç F, choose some Ω0 G ^ with F ç Ω 0 , and Fj u F 2 u · · · G ^ ΩΟ follows; thus we get stability under bounded countable union; the other properties of a local cr-ring are still easier to check; thus equality holds in (1). 2. Put Λ0 = {Fl u F 2 u · · · \Fl9 F2,... G ^ 0 0 } . Clearly ^ 0 is stable under countable unions, in particular finite unions. But it is also stable under differences: F l 5 Gl5 F 2 , G 2 , ... G« aoo

(y#H-y

G f AU * , fc

= U Πk

(Fn\Gk)

which belongs to ^ ° since f|fc (F„\Gk) G ^ 0 0 (η = 1, 2, ...) (we know by Remark 4.2.2 that a local σ-ring is stable under countable intersections). Thus 31° is a σ-ring. Since it contains a00 and clearly is contained in ^ ° ( ^ 0 0 ) , we have ^ ° ( ^ 0 0 ) = &°. The second equality in 7.2.1 now follows by the first entrance decomposition 1.3.3. 2.2 is now an easy exercise in case E e &°(&00). The general case will follow from 3.1. 3.1. Let a = {F|F G ^ 0 or Q\F e a0}. Clearly l ° ç | g # ( # ° ) . For 1, it suffices to prove that M is a σ-field. Ω G ^ is obvious. Let us now prove that «# is stable under differences. For E, F e a0, we haveE\F e <%° ^ B, (Ω\F)\F = Ω\(£ u F) G # s i n c e E u F e l 0 , £\(0\F) = £ n f e l ° ç | , (Ω\£)\(Ω^) = F\E G ^ 0 ç ai. Observe that here F\F, E\(Q\F) are even in ^ ° ; i.e., subtraction of members of a does not lead out of $°. Finally, let us prove that a is stable under countable unions:

46

I.

POSITIVE CONTENTS AND MEASURES

choose F 1 , F 2 , . . . e « ; if Fu F 2 , ...e<#°, then if, say Q\Fi G ^ 0 , then

^ υ ί

2

υ · · · £ ΐ

0

ς | ;

( Ω ^ ) n (Q\F 2 ) n · ·· = Π [· ·· ((0\F 1 )\F 2 )\ ·· -)\FJ; by our above observation, [· · · ((Q\F 1 )\F 2 )\ · · )\F„] G i%°\ since J>° is stable under countable intersections, Q\(FX u F 2 u ···) = ( Ω \ ^ ) n (Q\F 2 ) n ··· e # ° follows, hence Fl u F 2 u · · · e $. 3.2 follows from 3.1. 4 follows because every local σ-ring is a ring, every σ-ring is a local σ-ring, every σ-field is a σ-ring, every field is a ring, and every σ-field is a field. 5.1. Let us prove that ( J ^ o 6 l ^ ° is stable under countable unions. Choose Fu F 2 , ... G \J3Boel^° and assume F^e^tZ, F2e0H%€lL, .... Find a < e l containing # ? , »\, .... Then F x , F 2 , . . . e # ° , hence F i U F j U - e ^ g U « o e i # ° . The stability of 0<»o eI ^ ° under differences is still easier to prove (exercise). 5.2 follows from 1 and 5.1 (exercise). 4.8. Exercises 4.8.1. Let Ω Φ 0 be arbitrary and if = {{ω}|ω6Ω} the system of all one-point sets in Ω. Prove: 4.8.1.1.

0t{if) = %00{if) is the system of all finite subsets of Ω.

4.8.1.2. 3* (if) is the system of all sets ç Ω that are either finite or have a finite complement. 4.8.1.3.

^(if)

is the system of all (at most) countable subsets of Ω.

4.8.1.4. 0&(if) is the system of all sets ç Ω that are either countable or have a countable complement. 4.8.2. Let 0t be à ring in Ω φ 0 . Show that J ^ ) = {F|F e ^ or Ω ^ € St\ 4.8.3. Let Ω Φ 0 and if ç ^(Ω) be finite, say, if = {Fx, ..., F„}. Let Q 0 = F 1 u * " u F n . For every k = 1,..., n, let F° = Fk, Fj} = Q 0 \F fc . Prove that the nonempty sets among the sets of the form F\l n · · · n F*n (fij, . . . , £ B = 0 o r 1) form a partition of Ω 0 and that ^(5^) consists of all finite unions of atoms of this partition. Prove that 3F(if) is obtained by putting Ω 0 = Ω. 4.8.4. Let Ω / 0 and 0 φ F <^if ^0>(Ω) be such that every F G if is contained in some Ω 0 G 3Γ^ and ^~ is increasingly filtered : Ω^.,.,Ω,,ε^

=> Qj

for some Ω 0 G ^~. For every Ω 0 G ^", let:

H,çQo

4.

4.8.4.1.

σ-RINGS, LOCAL σ-RINGS, AND σ-FIELDS

47

^ Ω ο be the ring generated by { F | ^ 9 F ç Ω0}. Prove that

4.8.4.2. ^ Ω ο be the σ-field in Ω0 generated by {F\
=> Ω 1 , Ω 2 , . . . ^ Ω 0

for some Ω0 e ΖΓ. 4.8.4.3. Prove that a°(Sf) =[]Ωο^ ^ Ω ο · 4.8.4.4. Let ^ Ω ο be the σ-field in Ω generated by {F\ (Example 1.4), <^2

=

{[a, b[| - oo < a < b < oq},

^ 3

=

{[a, fc]|—oo < a
^ 4

=

^ 5

=

{]a, b[ | - oo < 0 < fr < oo}, {]a, b ] | - o o < a < 6 < oo},

^ 6

=

{]-00,fe[|fc€R},

.r7 = {]-ao,fc]|&eR}, ^10

{]fl, Oo[|tf6R}, = {[a, oo[|aeR}, = { F | F ç R, F compact},

^11

=

^12

=

{G|G ç R, G bounded and open}, { F | F ç R, F closed},

^13

=

{ G | G ç R, G open}.

^ 8

^

Prove: 4.8.6.1. ^00(^1):

9

=

(y2) = ^ 0 0 ( y 3 ) = ^ 00 (,^ 4 ) = ^ 0 0 ( t r 5 ) = ^ 0 0 ( ^ 1 0 ) = ^00(^u)·

The members of the local σ-ring described here in various fashions are called the bounded Borel sets of U.

48

I.

POSITIVE CONTENTS AND MEASURES

4.8.6.2. (/\, ..., V 1 3 all generate the same σ-ring, and this σ-ring is a σ-field .#(R). The members of this σ-field are called the Borel subsets of the real line R. 4.8.7. Carry over 4.8.6: 4.8.7.1. to R. The result Js the σ-field #(R) of all Borel subsets of R. Prove that #(R) = {E\E c R, £ n R is a Borel set in R}. 4.8.7.2. to Rn. The result is the σ-field @(Un) of all Borel subsets of R"(n = 1,2,...). 4.8.7.3. to C by an obvious identification of C with R2. The result is the σ-field <^(C) of all Borel subsets of C. 4.8.7.4. to C" by an obvious identification of Cn with R2w. The result is the σ-field of all Borel subsets of Cn. 4.8.8. Let n ^ 1 and Sfu ..., ifn be systems of subsets of R such that @(yk) = £(M) (k = 1, ..., n). Prove that ^(R n ) is generated by the system y = (J {Uk~ i x F x R"-fc-l \F e
4.9. Proposition. Let £f be a system of subsets of il φ 0 that is stable under finite intersections: Fl9...9Fne& 00

Let <%

ç ^

00

=> F1n--n

Fne6f.

( ^ ) be such that

(a) ^ ç l 0 0 (b) J' 00 is stable under proper differences, i.e., £, F e ^ 0 0 , £ ç F (c)

=> F\£ e ^ 0 0 ,

^ 0 0 is siafc/e wra/er (possibly empty) finite disjoint unions (thus in particular 0 e $00\ and under bounded countable disjoint unions. The latter means <%003FuF2,...ç:Fe<%00, =>

F

i + F2 + " - e ^

FjnFk=0 00

(j Φ k)

.

Then @00 = @00(5f). Proof Let &™n be the intersection of all set systems with the properties (a)-(c). Clearly ^°° n has properties (a)-(c) again, and it suffices to prove ^m?n = ^ 0 0 ( ^ ) · For this, it is sufficient to prove that @^m is stable against arbitrary finite intersections. Indeed, this allows us to write every difference F\E of two sets E, F e &™n as a proper difference F\(E n F) of two sets from JfJJi?n a n d every countable union of members of ^°° n a s a countable

5.

UNIQUENESS OF EXTENSION OF n-CONTENTS

49

disjoint union of other (namely smaller) members of ^°? η , by the first entrance decomposition (1.3.3). From properties (a)-(c) of .#°?n it will then follow that j C n is a local σ-ring containing y , and thus #£?η = jr 0 0 (^) as desired. The proof that @™n is finite intersection stable is achieved in two steps: (I) Choose any £ e ^ and consider £ g ° = {F\F e Λ°? η , £ n F e #£?„}. Since 5^ is finite intersection stable, we see S c ^ ° ° , i.e., ^£° satisfies (a). But it also satisfies (b). Indeed, if F, G e @°E°, F^G, then E n (G\F) = (F n G)\(F n F), being a proper difference of members of $™n, belongs to Λ°? η , hence G\F G ^g 0 . And @°E° satisfies (c). Indeed 0 e Ä°?n is obvious. If Fl9 F 2 , ... e ^°°, F,· nFk = 0(j* fc), then E n FU E n F 2 , ... e Λ ^ are pairwise disjoint and hence their union F n (Fx + F 2 + · · ·) belongs to ^^?n if o n ly finitely many of the Fk are # 0 o r else all Fk are contained in some F e@00. We conclude F! + F2+ - e ^ 0 in either case. Now @°E° 3 ^2,?n follows. Since F e 5^ was arbitrary, we see that Ee&>,

FeM^EnFe^l.

(II) Choose any F G <^°?n and define <#g° literally as above. Using the result of (I) we prove in the same fashion as above ^E°^^m?n> a n d thus finally F, F e ^ n ^>EnFe #£?„, as desired.

5. UNIQUENESS OF EXTENSION OF ^-CONTENTS

We now formulate the 5.1. Extension problem for contents. Let (Ω, 0t, m) be a content prespace. Does there exist a σ-content space (Ω, J*00, m00) that is an extension ο/(Ω, 0ί, m)? If yes, how many extensions exist? 5.2.

Discussion

5.2.1. Apparently, the σ-additivity of m is a necessary condition for a positive answer to the first question. 5.2.2. It seems reasonable to envisage not an arbitrary local σ-ring ^ 0 0 3 0t but ^ 0 0 ( ^ ) in the first line. This will in fact enable us to prove a uniqueness statement immediately. 5.3. Proposition. Ω Φ 0 . Then

Let m, m' be σ-contents on the local σ-ring @00 in @°0° = {F\F e ^ 0 0 , m(F) = m'(F)}

has the following properties:

(a)

06l§°;

50

I.

POSITIVE CONTENTS AND MEASURES

(b) ^o° is stable under proper differences; (c) ^o° is stable under finite disjoint unions and under bounded countable disjoint unions. If m and m' coincide on a set system 9* ^ &00 that is stable under finite intersections, then m and m' coincide on &00(9), in particular on &00 if 9 generates &00. Proof (a) m ( 0 ) = 0 = m'(0). (b) E,FG ^g°, E <= F, m(E) = m'{E\ m(F) = m'(F) imply m(F\E) = m(F) - m(E) = m'{F) - m'(E) = m'{F\E) ( c ) < , 3 F ! , F 2 , . . . Ç = F E < > , FjnFk=0 m(F2) = rri(F2) imply

(j Φ /c), m(Fl) = m ^ F j ,

Fj + F 2 + - - - e ^ g 0 , and m(F1 + F 2 + · · ·) = miFi) + m(F 2 ) + · · · = m'(F 1 ) + m'(F 2 ) + ··· = m'(Fl + F2 +

-)

by σ-additivity of m and m'. The proof that &Q° is stable against finite disjoint unions is still easier. The rest of the proof follows from Proposition 4.9. The reader should observe that this proposition implies its own special case for σ-fields: If m and m' coincide on a finite intersection stable set system 9 3 Ω, then they coincide on 0$(6f). As a consequence of Proposition 5.3, we get the following result concerning uniqueness of extensions: 5.4. Proposition. //(Ω, 0t, m) is a σ-content space, then there is at most one σ-content on &00(3$) extending m. The corresponding existence statement is also valid and will be proved in Chapter II (Theorem II.2.3 resp. Corollary II.2.4), and once more in Chapter III (Exercise 6.5.7). 5.5. Exercises 5.5.1. Let Ω τ* 0 . A system M of subsets of Ω is called an isotone class if it is stable under countable monotone unions, i.e., if Eu F 2 , ...eJi implies El ^ E2^

-

=>

Ê i U Ê j U - ' - e J

6.

THE HAHN-BANACH THEOREM

51

and an antitone class if it is stable under countable monotone intersections, i.e., if Εγ 3 E2 ^ ■ · · => Ei n E2 n · · ■ e Ji. An isotone and antitone class is also called a monotone class. It is clear what we understand by the monotone class Ji(Se) generated by a
A set ring is a σ-ring iff it is a monotone class. If $ is a ring in Ω, then 0$™{0t) = the antitone class generated

5.5.1.3.

If se ç ^(Ω) is finite intersection stable, then so is Μψ>).

5.5.2.

Prove Proposition 5.4 anew, using monotone classes.

6. THE HAHN-BANACH THEOREM

Having pushed forward the theory of contents to a certain degree, we shall now leave it for a while and study what we will call measures. Measures will be nothing but certain linear forms on certain real vector spaces, and measure theory will turn out to be an extension theory for these linear forms. Thus it seems to be feasible to prelude that theory with one of the most powerful extension theorems for linear forms, the Hahn-Banach theorem in its classical form. We present it together with some standard applications, one of which (Banach limits) will later on yield an important example of a measure. The proof of the Hahn-Banach theorem will also be given in its now classical form. It should be noted that the HahnBanach theorem can be incorporated into a theory of convexity in topological vector spaces, and proved by a different method in that context (see BOURBAKI [1], KÖTHE [1, 2], SCHAEFER [1]).

6.1. Theorem ( H A H N - B A N A C H ) . Let H be an arbitrary real vector space and p: H^U a so-called majorant function on H, i.e., a function such that p(f + 9)^p(f)

+ p(g)

ρ(α/) = αρ(/)

UgcH).

(0^«6f,

/ei/).

Let H0 be a linear subspace of H and m0 a linear form on H0 such that

m0(f) ^ p(f)

(f e H0).

52

I.

POSITIVE CONTENTS AND MEASURES

Then there is at least one linear form m on H extending m0, i.e., coincides with m0 on H0, such that

(/e//).

m(f)èp(f)

Proof. A pair (Hi9 m^) where Hx 2 H0 is a linear subspace of H and mi is a linear form on Hl that coincides with m0 on H0 and satisfies m 0 (/) ^ p(f)(f e i/i ) is called an admissible extension. If (//1? mi), (H2,m2) are admissible extensions, // χ ç // 2 and m2 coincides with mi on //^ we call (H2, m2) an extension of (Hu m j and we write (Hu γηχ) ^ (/J2, w2), and <\ίΗ1φ H2, in which case we speak of a proper extension. It is easy to see (exercise) that the set Σ of all admissible extensions, endowed with the partial ordering ^, satisfies the hypothesis of Zorn's lemma, hence there exists an admissible extension (Hu γηχ) that is maximal in the sense that there is no proper extension of it in Σ. We shall be through if we can deduce Hl = H from this. Assume ΗΧΦ H and choose some h2 e H\H1. For any/j, gx e Hu we have ^i(flfi) - Wi(/i) = mi(0i - / i )

^ P t o i - / i ) = P(tei+*2)-(/i+*2))

ipfei + M + pf-l/i + M) hence

-P(- (fi + h2)) - mi(/i) ^ p(gx + h2) - ηιχ(βι). If we put a = sup [-p(-(fi

+ h2)) - mi(/i)],

j? = inf [p{gx + h2) -

mM],

we have —oo 0 from ™2{h + ΑΛ2) = m^/i) + Ay ^ mii/O + AjS ^mi(/i) + A

«T

= P(fi + ^ 2 ) , and by a similar argument for λ < 0.

+

*'

)-(*)]

6.

THE HAHN-BANACH THEOREM

53

6.2. Example. Let |||| be a seminorm on the real vector space H9 i.e., a nonnegative real function on H satisfying

||α/|| = |α|· 11/11 (aeR,

f e H),

\\f + g\\ 1 \\f\\ + \\g\\

(f,geH).

Let m0 be a linear form on a linear subspace H0 of H such that |m 0 (/)| ^ ||/|| ( / G H0). Then there is a linear form m on H that extends m0 and satisfies |m(/)| ^ ||/|| ( / G /ί). Taking in particular any Q^f0eH9 H0 = {a/o I« e R} and m0(a/0) = a||/|| (a G R), we get m with m(f0) = \\f0\\. 6.3.

Exercises.

Let ||χ||αο = sup |x„| < oo

Γ = \x = {x09xl9...)\x09xl9...eU9

6.3.1. Prove that Z00 is a real linear space and HH^ is a norm on /°°, i.e., a seminorm satisfying flx^ = 0=>x = 0 ( = (0, 0, ...)). 6.3.2.

For every x = (x 0 , Xi, ...) G /°°, define |

r-l

p(x)= inf sup %aJxj+*\r=

\

1>αο» •••»«r-i ^ 0 , a 0 + ··· + αΓ_! = 1|.

Prove that this p satisfies the general requirements imposed in Theorem 6.1, (p(x + y) ^ p(x) + p(y) (x, ^ G /°°), ρ(αχ) = αρ(χ) (0 ^ a G R, X G Z00)). 6.3.3. Prove that the p() defined in (b) has the representation jr-l

p(x) = lim sup- £x;+„ 6.3.4. Let /conv = {x|x = (x 0 , xl9 . . . ) G /°°, lim,,-«, xn = Lim(x) exists}. Prove that Lim g p on /conv. 6.3.5. Let /Cesâro = {x|x = (x 0 , x l9 . . . ) G Z00, lim„(l/n) ^"=o */ = Lim^x) exists}. Prove that Lin^ g p on ZCesâro. 6.3.6.

Prove that for every x G /°°, ye /conv, p(x + y) = p(x) -f Lim(y)

6.3.7.

Prove that for every x G /°°, there is a linear form L on Z00 such that L(y) = Lim(y) L(x) = p(x)

(yelCOTV)

L(z)^p(z)

(ZG/ 0 0 ).

6.4. Example (Banach limits). Let Γ°, /conv, p, Lim be as in Exercise 6.3. A linear form L on Z satisfying (a)

L(x) ^ 0 (x = (x 0 , xl9

...)G

/»,

X0,

x l9 ... g; 0),

54

I.

POSITIVE CONTENTS AND MEASURES

(b) L((l, 1,...))=1, (c) L{(xl9 x2, ...)) = L((x0, x lf ...)) ((*o, xu · · ·) e /°°) is called a Banach limit. Let us prove that a linear form L on /°° is a Banach limit iff (b) holds and L(x) S p(x) (x e /°°). In fact, if L is a Banach limit, then L(\XQ , Xi, . . . ) ) = · · · = L(xr-1,

= L

\\

xr,

...))

Σχο+,·,- Σ*ι+;>···

r

j =0

' j =0

II

=

S U

P;

Σχη+]

n^O ' j = 0

((x0, Xi,...) e /°°) follows from (a)-(c) and hence L(x) ^ p(x) (x e /°°) holds by Exercise 6.3.3. In order to prove the converse, we show that p((xi, x 2 , ...) — (x 0 , Xi, )) = 0 ((xo, Xi, . . . ) G '°°)· I n ^act w e have p{(xu x 2 , . . . ) - (x 0 ,Xi, .·.)) jr-l

{

= lim sup - Σ (xj+i+n - *;+«) = Hm sup - (xr+„ - x„) r-*co n ^ O

r

^ljm^MU r-»oo

r-+oo n ^ 0

j=0

'

= 0

r

(x = (x 0 ,x 1 ,...)e/°°).

Thus L(x) ^ p(x) (x G /°°) implies L((x1? x 2 , ...)) - L((x0, x i? ...)) ύ 0 and thus (multiply by —1) L((xl9 x 2 , ...)) —L(x0, x l5 . . . ) ) ^ 0 , hence (c). In order to prove (a) we need only observe L(x)^p(x)^0

(x = (x 0 ,x 1 ,...)e/ QO ,

x0,xl9...£0)

and multiply by — 1. This was to be proved, and Exercise 6.3.7 shows the existence of Banach limits; Exercises 6.3.4 and 5 imply that Banach limits are extensions of the usual limits and Cesâro limits. 7. ELEMENTARY DOMAINS

For any nonempty set Ω, the system IR" of all real functions on Ω is a vector lattice if we define the linear and lattice operations pointwise. Let us list the most important among the so-called finite vector lattice operations, and some of their relations (v stands for "maximum "and Λ for"minimum" as usual): /+ = / v 0 = Uf + I / 1 ) /_ = - ( / Λ θ ) = ( - / ) ν Ο = ( - / ) + = i ( - / + I/1)

(positive part); (negative part);

7.

ELEMENTARY DOMAINS

l/l = / v ( - / ) = f++f.=f+ + (-f)+ / v g = / + (g - f)+ (supremum); / Λ g = / - (/ - g)+ (infimum);

55

(modulus);

f = U-f-. Observe that the operations (·)+, (·)_ , | · | , ν , Λ make sense also in the system W1 of all extended real-valued functions (R = (R u { + oo, — 00} with the usual conventions about ordering), while linear operations suffer under the problem of avoiding 00 — 00, and are thus not always defined in Mn. Some relations like/ = /+—/__ make sense nevertheless. It is these pointwise defined operations in Un or (RQ that are tacitly meant whenever we speak of lattices or vector lattices of (U- or R-valued functions. For any £ ç l " w e define S+ = {/10 ^ / e S}. For any / e I " , the set supp(f) =

{œ\f(œ)*0}

is called the support or carrier off. From the above equalities we may deduce: 7.1. Proposition. A linear subspace ofUa is a vector lattice iff it is stable under one of the operations ()+ , | | . 7.2. Definition. Let Ω Φ 0 . A vector lattice S ç Ua is called an elementary domain on (the basic set) Ω if it satisfies 7.2.1. Stone's Axiom: fe£=>fA(xeE(oi = const > 0). In this context, the members of S are also called elementary functions.

7.3. Remark. Every vector lattice ê ç R" contains the constant 0. If it contains any constant Φ 0, then it contains all real constants and is an elementary domain. We shall encounter natural examples of elementary domains that contain no constant except 0. Stone's axiom has its name after Marshall Stone (b. 1903). 7.4. Example. Let Ω Φ 0 be arbitrary. Then &* is an elementary domain on Ω that contains all constants. {flfeâP, supp(/) is finite} is an elementary domain that contains no constant φ 0 unless Ω is finite. {f\f e R", supp(/) is at most countable} is an elementary domain that

56

I.

POSITIVE CONTENTS AND MEASURES

contains no constant φ 0 unless Ω is at most countable. The system of all bounded real functions on Ω is an elementary domain that contains all constants. 7.5. Example. Let Ω = R. Then g = < - / f a - * >

(x

_ ak_x)

( ^ ix^ak,

k= 1 , . . . , n).

For topologists we include here 7.7. Example. Let Ω φ 0 be a topological space. Then the following linear subspaces of R" are elementary domains: 7.7.1. the system ^(Ω, R) = #(Ω) of all continuous real functions on Ω; 7.7.2. the system ^ ( Ω , R) = ^ ( Ω ) of all bounded real functions on Ω; 7.7.3. the system #°(Ω, R) = ^°(Ω) of all continuous real functions on Ω vanishing at infinity (an / e R" is said to vanish at infinity if for every ε > 0 there is a compact K ç Q such that |/(ω)| < ε (ω e Ω\Χ)). 7.7.4. the system # 00 (Ω, R) = {/1/ e #(Ω), supp(/) is compact} = # 00 (Ω) (the bar means closure) of all continuous real functions on Ω with a conditionally compact support. 7.7.5. We have «if (ft R) 3
Example.

Let 0t be a set ring in Ω Φ 0 . Define

ί(Λ) = { αι 1 Γι + ··· + α Η 1^|ϋ^ l , a 1 , . . . , a l l e R , F 1 , . . . , F l l € d f } . The members of ${$) are called the step functions for $. It is obvious

7.

ELEMENTARY DOMAINS

57

that S{M) is a linear subspace of Rn. (2)

/ = /?il ß l +

·+&1Β,

is called a representation of / G e(ß) in disjoint form if B1? ..., Bre0t are pairwise disjoint. Let us show that every / e &{$) has a representation in disjoint form. Clearly, every indicator function 1F of some F G 3t is a step function for 0t given in disjoint form. We are through as soon as we can prove that every linear combination of functions given in disjoint form can be represented in disjoint form. Let thus f = ßiUl+" + ßrhr 9 = yilcl + '- + yslcs be two functions from E(R) given in disjoint form. Putting B r+ i = (C1 + "- + Ce)\(B1 + -" + Br)> Cs+! = (*! + ··· + Br)\(Ci + '·· + Cs),

ßr+l

= 0 = ys+1

we see that we may assume without loss of generality B1 + · + B r = C l + · · + C S . Put now Djk = Bj n Q , j3jk = jfy, yjk = yfc (/ = 1, ..., r,fe= 1, ..., s). Then clearly r

s

1

f= Σ LA* **. j=lfc=l

#

=

r

s

Σ Ex/*1**

j=l

fc=l

are representations of/, # in disjoint form. Now we get, for any reals λ, μ, the following representations of Xf + μ# in disjoint form: r

s

tf + w = Σ Σ Wfo + W,*)1D, · Now that we know that every / G
shows | / | G e(ß) and from Proposition 7.1 e(ß) is a vector lattice. r

f A0t= Σ()3;Λα)1β. for every real constant a > 0 shows that ê{$) also satisfies Stone's axiom, hence is an elementary domain.

58

I.

POSITIVE CONTENTS AND MEASURES

8. MEASURES ON ELEMENTARY DOMAINS

8.1. Definition. Let ê be an elementary domain on Ο.Φ 0. A real function m. $ -*U on $ is said to be: 8.1.1.

8.1.2. form and

linear or a real linear form if m(*ifi + · · ' + «„/„) = *Mf\) + ··· + ocnm(fn) (n^l, <*!,...,«„€ R, Λ /Βεί). a positive linear form or a (positive) measure on S if it is a linear

m(f)^0 8.1.3.

σ-continuous (at 0) if lim m(fk) = 0

Λ-αο

8.1.4. form on 8.1.5. &ofê+

(OufeS).

(S 3 fx ^ f2 :> · · · -► 0 pointwise).

a (positive) σ-measure on ê if it is a σ-continuous positive linear S. τ-continuous (at 0) iffor every decreasingly filtered nonempty subset with inf f(œ) = 0

(ω

G

Ω)

the relation lim m(f) = 0 holds, i.e., if for every ε > 0 f here is an / 0 G 3F such that f0^ f G 3F implies | m ( / ) | < ε. 8.1.6. a (positive) τ-measure on S if it is a τ-continuous positive form on S. 8.1.7. If m is a (positive) measure on the elementary domain S on Ο.Φ 0, then the triple (Ω, ê, m) is α/so ca/fed a measure space and properties of Ω, ê, or m are (sometimes imprecisely) also attributed to the triple. In particular, it is clear what a σ- or τ-measure space is. 8.1.8.

If m is a positive measure on S, we call the extended real number ||m||=sup{m(/)|/G(f,0^/gl}^0

the total mass or total variation of m. If ||m|| < oo, we call m a bounded (positive) measure; if ||m|| = 1, we call m normalized. / / the constant 1 belongs to S and m(l) = 1, then we call m a probability measure and (Ω, S, m) a probability (measure) space. / / ||m|| = oo, we call m an unbounded (positive) measure.

8.

8.2.

MEASURES ON ELEMENTARY DOMAINS

59

Remarks

8.2.1. The above definition uses some conventions and concepts which should be generally known. In order to ensure complete understanding, let us recall the general conventions about filtered sets and limits. For every set M φ 0 endowed with a partial ordering ^, a subset F of M is called increasinglyfilteredif «for any n ^ 1 and xl5 ..., X„GF, there is an x G F with Xj ^ x, ..., x„ ^ x. F ç M is called decreasinglyfilteredif for any M ^ 1 and xl5 ..., x„ e F, there is an x G F with x ^ x l9 ..., x ^ x„. If F ç M is nonempty and increasingly filtered, if m: M->R'is any real function and a a real number, we write limF m(x) = a if for every ε > 0, there is a x 0 G F such that F 9 x ^ x0 implies \m(x) — α | < ε. An analogous convention is made for decreasingly filtered sets, and a generalization to filters as well as mappings m of M into any topological space obviously makes sense. 8.2.2. A real linear form m on a vector lattice ê ^ Ua (it could be an abstract one as well) is positive iff it is isotone, i.e., iff, /, g e S, fè g=>m(f) S m(g) (form g -f). 8.2.3. Clearly a probability measure is normalized. We shall, of course, speak of a σ-probability (τ-probability) space if we have a probability measure that is σ-continuous (τ-continuous). 8.2.4. A positive measure m on an elementary domain S is σ-continuous at 0 iff it is σ-yî/rer-continuous at 0, i.e., iff for every countable decreasingly filtered nonempty subset m(fn) - m(f) (form /„ - / ) . An analogous statement is evident for σ-filter-continuity and for τ-continuity. 8.2.6. Clearly τ-continuity implies σ-continuity. 8.2.7. The real linear forms on an elementary domain S form a linear subspace of U6. The positive measures on S form a set that is stable under

60

I.

POSITIVE CONTENTS AND MEASURES

finite linear combinations with nonegative real coefficients. The probability measures on an S containing all constants form a convex set. 8.3. Example. Let S be an elementary domain on Ω Φ 0 . Choose some points ωΐ9 ..., ωη e Ω and some reals al5 ..., a„. Then

(3)

«(/·)= Σ«*/Κ) fc=l

defines a τ-continuous linear form on S. If a1? ..., a„ ^ 0, then m is a (positive) τ-measure; in this case it is bounded with total mass OLX + ··· + απ. In case n = 1, ocj = 1, it is called the Dirac measure or pomi mass one at ωχ and denoted by <5ωι (some authors prefer εωι). In due course, the m defined by (3) can be written m = α^ ω ι + · · · + (χηδωη. All this carries easily over to countably many points ω ΐ9 ω2, ... G Ω if we suppose |ax | -h |a21 -l· * * * ) φ /(η), Assume further that for every ω G Ω, there is a g G ^ with #(ω) # 0. Prove that then for any ω,rçG Ω with ω φ ^y, there is even some Λ G S such that /ζ(ω) # 0 = h(n), and αι^ωι + ··· + α„<5ωη = β ^ + '" + βΛ9 in case 0£ {αΐ5 ..., α„, 01? ..., &}, ω} φ œk (j φ /c), ηί Φ ηι (i φ I) implies η = s and, after a suitable renumbering of ηΐ9 ..., ηη, <χ.γ = j?1? ..., α„ = β„, 8.4.2. Let Ω = R, 0 and m(f) = 0; show that the set of all empty points in R is open and Φ R; for any nonempty points ω φ η form 0 g /, g e # 00 (R, R), such that /(ω), #(>?) > 0, /g = 0 in order to get a contradiction). 8.5. Example. Let Ω = R, g = # 00 (R, R), as in Example 7.5. Then every measure m on S (in particular the Riemann integral defined by m(f)= i f(x)dx) is σ- and even τ-continuous. In fact, if &+B fi^ f2^:'"->0 pointwise, then for every ε > 0, the sets Fn = {fn > ε) (w = 1» 2, ...) are compact and satisfy Fx Ώ. F2 2 · · ·,

8.

MEASURES ON ELEMENTARY DOMAINS

61

Fx n F2 n · · · = 0 . A well-known topological argument shows the existence of some n0 with F„0 = 0 , i.e.,/, ^ ε (n ^ n0). We may now easily find an f0eS with l supp(/l) ύ fo and conclude m(/n) g m(e/0) = em(/0) (" ^ "o)· Since ε > 0 was arbitrary, lim,,^ m(fH) = 0 follows, τ-continuity follows in a similar way (exercise), and it is easy to carry over the example to Rn and S = #00(Rn, R) (exercise). This example can be subsumed under Theorem 10.2. Measures m: <
((x0, * i , . . ) e £).

Letnow/„ = (x 0 ,Xi,.-.) where x 0 = · · = x„ = 0andx„ + 1 =xn + 2 = '"= 1. Clearly S ^ fx ^ f2 ^ · · · -► 0 pointwise. On the other hand, normalization plus shift invariance imply m(fn) = m((l 1,...))=1

(n=l,2,...).

Thus m is not σ-continuous. 8.7. Example. Let (Ω, 0t, m) be a content prespace and e(ß) = {oillFl + ··· + 0Ln \Fn\n ^ 1, a 1? ..., a„ e R, F x ,..., F„e R} be the elementary domain of all step functions for R (Example 7.8). Then there is a unique positive measure (again called) m on $($) such that (4)

m(lF) = m(F)

(F e @)

(the left m denotes the measure and the right m denotes the content). In fact, as shown in Example 7.8, every/ e ê{0t) has a representation (5)

f = ßilB,+

-+ßrlB,

in disjoint form. A representation (5) of a n / e ê{$) is called standard if Bl9 ..., Br Φ 0 , Bj n Bk= 0 (j^ k), /?!, ..., ßr Φ 0, ßj Φ ßk (j Φ k). It is clear that every 0Φ f e £{&) has a unique standard representation: {ßu ..., ßr] = /(Ω)\{0}, Bj = {/ = ßj} (j = 1, ..., r). It can be obtained from any representation in disjoint form by discarding zero summands and uniting some of the sets

62

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involved. Put m(0) = 0 and (6)

m(f) = ßMFi)

+ -- +

ßMFr)

if / Φ 0, and (5) is a standard representation of / This defines an m: ê{$)^> U that clearly satisfies m(f) ^ 0 (0 ^ / e ê{$)\ Let us now prove that it is a linear form. For this we first observe that (6) holds true also if (5) is only assumed to be in disjoint form (exercise). But now, if we h a v e / g e ${$), we know from Example 7.8 that there are representations (5) o f / a n d (?)

9 = ΊΛΒ, + ··· + ινΐΒΓ

in disjoint form with the same sets B} in (5) and (7) (they were called Djk in Example 7.8). Now for any reals λ, μ, we find that ¥ + W = (λβχ + W i ) l e i + ' ■ · + (Wr + Wr)l*r is a representation in disjoint form, hence m(¥ + W) = (A/Î! + W i M ^ i ) + ' · · + (λβ, + μγ,)(Β,) = hn(f)

+ μ™(ο)

as desired, m is thus a positive measure on S($), and it is clear that it satisfies (4) and is the only one satisfying (4). Clearly (6) holds even if (5) is not in disjoint form. We call it the measure on 8{0t) derived from the content m on R, and consequently (Ω, S(β), m) the measure space derived from the content prespace (Ω, 0t, m). 8.8. Convention. Properties, statements, etc. based on a content prespace (Ω, 0t, m) will bê attributed, possibly with appropriate modifications of language, to the derived measure space (Ω, S[ß), m) and vice versa. 8.9. Proposition. Let (Ω, e(ß\ m) be the measure space derived from the content prespace (Ω, 0ί, m). Then the measure m is σ-continuous iff the content m is σ-additive. Proof. " if" : let fn = J > s t ßnj \Bn. e S (a) be such that / i £ / 2 £ · · · - > 0 pointwise. We may assume/^ f2, ... / 0 and all representations standard. With M = s u p ^ ; ^ ßl} we have 0 ^ / „ ^fx ^ M (n = 1, 2, ...). Choose any ε > 0 and put En = {fn ^ ε} = [jßnj^ Bnj. Clearly El9 E2, . . . 6 « , £ χ 3 £ 2 3 · · · , £ 1 η £ 2 η · · · = 0 . By Proposition 2.2 we can conclude m(En) - 0. Now 0 ^ /„ ^ ε 1 / ι > 0 + M\En shows

O^mifJ^emttf^OV

+ MmiEj.

9.

RIEMANN (EUDOXOS) EXTENSION OF POSITIVE MEASURES

63

Since ε > 0 was arbitrary, lim,,.^ m(fn) = 0 follows. "only if": Let 0t B EX Ώ E2 2 ■ ■ ■, £i n E2 n · · · = 0 . By Proposition 2.2 it suffices to prove m(En)-+0. But 1E„-»0 pointwise, hence m(En) = m(lEn) -► 0 follows from (4) and the σ-continuity of m. 8.10.

Exercises

8.10.1. Let A Φ 0 be afiniteset and Ω = 4 x 4 x · · · the Bernoulli space with the alphabet A. A function / e R" is said to be of order r ^ 0 if there is a function fr e R^r+1 such that /(COOCÜ! · · · ) = / Γ ( ω 0 · · · ω Γ )

(ω = ω 0 ω !

· e Ω).

Let Er = {f\fe R n ,/is of order r} (r = 0, 1, ...). Show that every Er is an elementary domain on Ω, that £ 0 — £i ^ * ' * an d E = E0 u Ex u · · · is an elementary domain on Ω, too. Prove that every positive measure on ê is τ-continuous. (Hint: Use Example 2.5.) Prove that for every positive measure m on ê the measure space (Ω, 8, m) is derived from a uniquely determined content prespace (Ω, #, m) where # is defined according to Example 1.6. Prove anew that every content on # is a σ-content. 8.10.2. Let Ω Φ 0 and ê ç (RQ an elementary domain on Ω. Assume that every/ G S is bounded. Let p: S -► R be defined by p(f) = sup weiî /(ω). Prove that p is a majorant function and apply Theorem 6.1 (Hahn-Banach) in order to prove the following: Let S0 be a linear subspace of S and m0:$0-^>Ube linear and such that/0 e S0 =>rn0(f0) ;g p(f0)\ then there is a positive measure m: m(/) ^ 0.) We remark that this exercise comprises the following special case: S contains all constants and S0 consists of all constants, m0 carries every constant into itself. There is still an important class of examples of measure spaces to be described—those with a topological space Ω as a basic set and ^(Ω, U) or #00(Ω, R) as an elementary domain. We postpone them until Section 10 and continue now by presenting comparatively simple aspects of the extension problem for measures.

9.

RIEMANN

(EUDOXOS)

EXTENSION OF POSITIVE MEASURES

The examples of measure spaces presented so far convey themselves already ample suggestions for extensions of measures from the elementary domain on which they are given to a larger domain.

64

I.

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Let, e.g., (R, ®, m) be the dyadic content prespace on R (Examples 1.10, 2.4) and (R, (f(^), m) the derived measure space. Apparently ê(2) is nothing but the space of all real functions on R assuming finitely many real constant values on finitely many dyadic intervals and vanishing elsewhere, and m is the usual elementary (pre-Riemannian, so to speak) integral on S(ß). Every course in calculus provides enough material which makes it desirable to "integrate," i.e., to include into an extended domain for m, functions which are not yet in e{ß\ such as functions that vanish outside some interval and coincide with a polynomial, a piecewise linear function, or a step function with nondyadic intervals of constancy, or even a suitable rational function inside, not to mention functions like 1Q · 1[0, i] (Q = the rationals) which might look pathological at first sight. In this subsection we present a simple method of extension of measures which is analogous to that given for contents in Section 3 and might well be named after Eudoxos (ca. 408-355 B.C.) again but is usually named after Riemann (1826-1866). We supplement it by considerations about extension by uniform approximation. 9.1. Definition. Let (Ω, S, m) be a measure space. A function g e R" is called Riemann integrable for (Ω, $, m) or, briefly, for m, if for every ε > 0 there are f h e ê such that fuguK

m(h-f)
or, equivalently (obey the rule inf 0 = oo, sup 0 = — oo) — oo <

sup m(f) =

inf m(h) < oo.

Let ${m) denote the system of all Riemann integrable functions for (Ω, S, m). 9.2. Theorem. Let (Ω, ê, m) be a measure space and i(m) the system of the corresponding Riemann measurable functions. Then: 9.2.1.

£{m)=>Sand m(g)=

sup m(f)

(flfe
defines an extension (again denoted by m) of m from S to ${m). It is called the Riemann integral for m. 9.2.2. (Ω, ${m), m) is a measure space again. It is a σ-measure space iff(il,S,m)is. 9.2.3. Every positive measure on ${m) that coincides with m on E coincides with the m given in 1. (S{m)\m) = S(m). Proof. 1 is obvious: For g e ê, t a k e / = g = h. 2. Let us first prove that ê{m) is an elementary domain. For this, let us

9.

RIEMANN (EUDOXOS) EXTENSION OF POSITIVE MEASURES

65

be content to prove its stability under the passage g-> g+ = g v 0 to positive parts; stability under linear operations can be proved in a similar, and then obvious, way; and this will do it after Proposition 7.1. Now, if g G S(m) and ε > 0, choose f, he $ with f^ g ^ h, m(h —f) < ε. Then clearly ê a f+ ^ g+ ^ h+ G 8, h + - f+ ^ h - f, and thus w(fi+-/+)gm(fi-/)U is linear again. Let us first prove additivity: For any g, g' G S(m) we have m(g + g') =

sup 6 af^g

^

m(f) + g'

sup

m(f + / ' )

= sup m(f)+ Ssf^g

sup

m(f')

Ssf'^g'

= m(g) + m(g'). Now the definition of Riemann integrability, m(g) and m(g') can be computed as infima as well. The corresponding estimates yield m(g + g') ^ m(g) + m(g') and thus the desired equality

Ms + #') = w (#) + w (#') fo» 9' e ^)· The proof of m(ocg) = am(#)

(g G S,

a G (R)

is still simpler (exercise; distinguish the cases α ^ 0 and a < 0). Thus m: &(m) -► R is linear. It is also positive: For g ^ 0, choose/ ^ 0 in S. Let us finally prove that m: IR is. Let ê(m) s gt ^ # 2 = '" -^0 pointwise. For a given ε > 0, choose f\,f2, ...e$ such that 0 ^ / ; ^ ft, w ( / i ) £ m( ft ) - ε/2< and put fk = f\ A ··· Λ / ; (* = 1,2,...). Clearly fuf2,...

G*

and

0 £ / ; - Λ ύ (gk-i - fk-i) + '·· + (fli - / i )

and thus m(/;) ^ m(/fc) + ε/2*~* + · · · + ε/2 ^ m(fk) + ε and hence m(/fc) ^ m(öfk) — 2ε (fc = 1, 2, ...). But we have fk-*0 pointwise, thus lim m(gk) ^ 2ε, and finally m(gk)-+0 since ε > 0 was arbitrary. The rest is obvious. 3 is an easy exercise. 9.3.

Definition.

For any measure space (Ω, S, m) the measure space

66

I.

POSITIVE CONTENTS AND MEASURES

(Ω,
<£(**)= 1

(/c = 1,2,...)

but m(g\) + m(g'2) + · · · < 1. Put gk = g\ v · · · v g'k (k = 1,2,...), g = lim gk. Prove that gi(S(9)\m). (Hint: If g<,he&(®\ then f i ^ l ] 0 t l I , thus g 6 (S(Q))\m) would imply m(g) = 1; bring this to a contradiction with Σm(g,k) < 1.) 9.4.3. Let (R, (£(@)\m), m) be the σ-measure space defined according to 9.4.1. Prove that this m is not τ-continuous on (S(Q)))(my (Hint: Consider the decreasingly filtered system F of all functions 1F where F = [0, 1] minus a finite number of points.) We supplement the above sketch of Riemann integrability by some convenient results about uniform approximation (which has, of course, already played a key role in Exercise 9.4.1). We begin with a convenient special case in which we can dispose of constant functions. 9.5.

Exercises

9.5.1. Let (Ω, S, m) be a measure space such that S contains all constants. Show that E{m) is closed under uniform approximation. (Hint: Use (${m)\m) = $(m) and the fact that uniform approximation makes possible the simple "squeezing-in" procedure which defines Riemann integrability.) 9.5.2. Let A φ 0 be a finite set and ê the elementary domain of all functions of finite order on the compact metric space Ω = A x A x · · · (Exercise 8.10). Prove that ^(A x A x ···) is the uniform closure of ê, and thus in particular S{m) 3 <€(Α x A x · · ) for every positive measure m on S.

10.

MEASURE SPACES OVER TOPOLOGICAL SPACES

67

A general result on uniform approximation is given by: 9.6. Proposition. Let (Ω, S, m) be a measure space and $' an elementary $', there is domain of bounded functions on Ω such that for every f'eS'u anO^heS with {/' Φ 0} ç {h ^ 1}. Assume further that δ' is contained in the closure of S with respect to uniform approximation, then ê' ç S(m). Proof For any given g' e 8' find some 0 ^ h0 e δ such that {gf φ 0} ç {h0 ^ 1} and then some 0 ^ hl e S with {h? Φ 0} £ {/h ^ 1}. Since #' is bounded, we may (after multiplication of fi0 with some constant ^ 1) assume \o[\ ^ Jz0. Let M be a constant ^ ft0, ε > 0, and g0 e S such that \Q' — θο\ < ε uniformly on Ω. Replacing g0 by (g0 A h0) v ( — hQ\ we may assume \g0| ^ h0. Put now / = g0 — sh0, h = g0 + eh0. Clearly, S3 f ^g^heS a n d O ^ / i - / ^ 2ε/ζ0 ^ 2εΜΛ1. We conclude

Since ε > 0 was arbitrary, g e £{m) follows.

10. MEASURE SPACES OVER TOPOLOGICAL SPACES

The importance of measure spaces whose basic set Ω is a topological space and whose elementary domain is ^ ( Ω ) = #5(Ω, R), or ^ 00 (Ω) = # 00 (Ω, R) is a sufficient reason to devote a whole section to them. The reader may consider Example 7.7 and 8.5 and Exercises 8.10, 9.5.2 as a prelude to it. He is now supposed to be informed about the basic concepts and results of general (point set) topology. 10.1. Theorem. Let (Ω, ΖΓ\ Ωφ 0 be a compact topological space. Then every positive measure m on the elementary domain #(Ω, R) is τ-continuous. Such measures are also called Radon measures in Ω. Proof We essentially use Dini's theorem and observe that <<ί(Ω, R) contains all real constants. Let 0 = 0, the family ({/ ^ ε})/ € ^ consists of compact sets, is decreasingly filtered, and has an empty intersection. A well-known elementary result in topology (actually not more than the definition of the compactness of Ω applied to the open covering ({/ < e})/eir of Ω) says that there is an / e 3F for which {/ ^ ε} is empty, i.e., 0 ^ / < e, and consequently 0 ^ m(f) ^ em(l). Since ε > 0 was arbitrary, we find inf /eir m(f) = 0 = lim^ m(f), and thus the τ-continuity of m.

68

I.

POSITIVE CONTENTS AND MEASURES

In more general situations we must try to get some substitute for the constants in #(Ω, R). 10.2. Theorem. Let (Ω, 3~), ΩΦ 0 be a locally compact Hausdorff space. Then every positive measure m on the elementary domain <<ί00(Ω, R) is τ-continuous. Such measures are also called Radon measures in Ω. Proof Let 0 φ & ç # 0 0 (Ω, R) be decreasingly filtered with pointwise infimum 0. Choose any f0 e ^ and define F0 = {f\F s f ^ f0}. Clearly inf /ei r 0 f(œ) = 0 (ω e Ω) again and inf /ei r 0 m(f) = inf /eJ r m(f). Thus we may henceforth assume / ^ f0 ( / e &) for some f0 G ^. Consider the compact set K = {f0> 0}. For every ε > 0, the Dini-type argument in the proof of Theorem 10.1 yields s o m e / e & w i t h / ^ ε. Basic results in the theory of locally compact Hausdorff spaces (they go with the name of Urysohn) yield the existence of a n / ! e ^ 0 0 (Ω, R) such that 1K ^ fv Now we may conclude/ g ε/\ for our above/ and this implies 0 ^ m(f) ^ sm(fl). Since/j is fixed independently of ε and ε > 0 is arbitrary, inf /6jr m(f) = lim^ m(f) = 0 follows. This theorem applies, e.g., to Ω = Rn and m = the usual Riemann integral as a positive measure on ^ 00 (R n , R). 10.3. Definition. Let (Ω, 9~\ Ωφ 0 be a topological space. A linear form m on an elementary domain S contained in the space # b (Q, R) of all bounded continuous real functions on Ω is called tight iffor every ε > 0 there is a compact K ç Ω and a δ > 0 such that fe*,\f\£l,\f(a>)\
(a>eK)

=>

|m(/)|
If m is a tight (positive) measure, then the measure space (Ω, 0, we choose a compact K c Ω and a δ > 0 according to the definition of tightness. Put Kf = {ω|ω e Ω, / ( ω ) ^ <5} ( / G F). Then (Kf)feJ? is a decreasingly filtered family of compact subsets of X, with an

10.

MEASURE SPACES OVER TOPOLOGICAL SPACES

69

empty intersection. It follows that there is some/i e & for which Kf = 0 , i.e., |Λ(ω)| < δ(ω€ K), and consequently | m ( / ) | < e (& af^f^ which proves the desired statement. Next comes a theorem on the preservation of tightness under a certain approximation procedure. We begin with a technical observation: 10.5.

Remark.

Define generally for any Ω Φ 0 ,

ll/ll = sup l/MI

(/eR°)

CO 6 Ω

and observe that for any /, g e R" and any constant Ai G R such that < M we have \f-[(gAM)w(-M)]\^\f-g\. In particular, if & ç R" is an elementary domain and / G S satisfies H/ll g M, then for every gsS there is a g' e S with ||gr'|| g M and \f-9'\^\f-Ql 10.6. Theorem (HILDENBRAND [1]). Let (Ω, $~), ΩΦ 0 be a topological space. For every h e # b (Q, R), 0 < M < o o , K ç Q compact, δ>0 define V{h; M, Κ,δ) = { / | / G <^(Ω, R), ll/ll ^ M, | / ( ω ) - Λ(ω)| < δ

(ωεΚ)}.

b

Let 5^ be the topology in %> (Q, R) whose basis at h consists of all sets U(h; Ai, K, (5) with M = ||ft|| + 1. Let m be a tight linear form on an elementary domain S ^ ^ ( Ω , R). Then there is a unique linear form m on the ^-closure S of ê such that m is continuous for the restriction of 9" to S and coincides with m on S,m is positive if m is, and is tight. If S separates the points of Ω (i.e., if for any ω, η G Ω with ωΦη, there is an f se with f(co) φ /(^)) and contains all constants, then ê = «*(Ω, R). Proof. Going into the definition of tightness with a multiplicative constant > 0, we see that for any ε > 0, M > 0, there is a compact K ç Q and a δ > 0 such that / G ^(0; Ai, K, δ) η 8

=>

|m(/)|
This implies that for any ε > 0, M > 0, there is a compact K' ç Ω and a <5' > 0 such that for any h G «*(Ω, R) with ||ft|| ^ Ai / of G #(Λ; Ai, Κ', ô') n g

=> f-ge

« ( 0 ; 2Ai, K', 25')

=> Μ/)-"Κβ)| <«· By mere routine we may now prove: ê is an elementary domain again; for every heS with \\h\\ g Ai all W(h; Ai, K, δ) n S are nonempty; for

70

I.

POSITIVE CONTENTS AND MEASURES

every h e S, there is a unique real m(h) such that for ||ft|| ^ M / G 1/(Λ; M, Κ', ä ' ) n i

=>

|m(/) - m(fc)| < ε

if K\ δ' are chosen as above; m is a linear form on S, positive if m is. Let us now prove that m is tight. For this, choose an ε > 0 and then a compact K ç Q and a δ > 0 such that /e*(0;l,K,3)n* Put now M = l and choose K\ ||Λ|| ^ 1, we have /e«(Ä;l,K\i')n
=>

|m(/)|
δ' as above: For any he S with =>

|m(/) - m(/i)| < ε.

Then for any /z G ^(0; 1, K u &', \ min[<>, ό']) η 0, K ç Ω compact, R G M ^ ||fc|| + 1, choose fee according to the Stone-Weierstrass theorem such that |/'(ω) - Λ(ω)| < δ (ω G X). By 10.5 the function/ = ( / Λ M) v ( - M ) is in U(h; M, K, <5), as was to be proved.

11. THE EXTENSION PROBLEM FOR MEASURE SPACES

Exercise 9.4.2 more or less indicates the direction that has, by mathematical experience, turned out to be the correct one for a formulation of the 11.1. Extension problem for measure spaces. Let (Ω, £, m) be a measure space. Does there exist a measure space (Ω, if, m) such that S ç S£, m: i f -► R is an extension of m: ê -► R such that the following statement holds: Monotone convergence theorem: / / 0 ^ hx ^ ft2 ^ *·· G Jä?\ lim,,^ m(/z„) < oo, fJien ί/ie pointwise lim„_+Q0 Ji„ = h is in ï£ again ana we have m(h) = lim m(hn). n-*ao

11.2.

11.2.1.

Remarks

From the monotone convergence theorem above there follows

11.

THE EXTENSION PROBLEM FOR MEASURE SPACES

71

immediately a "downward version" of it, which is nothing but the σcontinuity of m: i? -► R (exercise). This implies, of course, the σ-continuity of m: 8 -> R. Thus the extension problem should a priori be considered for σ-measure spaces (Ω,
lim m(hn)= £ ^ = l < o o ;

n-oo

fc=l

2

but limn_00 /z„(0) = oo. This indicates that we have to admit functions with values + oo if we want to obtain the monotone convergence theorem as formulated in 11.1. But with functions attaining values in R = R u {oo, — oo} we run into difficulties when adding or subtracting them, since oo — oo cannot be reasonably defined. This difficulty is one of the raisons d'être of the so-called nullset business which the reader will encounter in the course of the extension process carried out in Chapter III and which is a characteristic feature of all modern integration (extension) theories; it runs, roughly speaking, as follows: we admit functions with values in R to i f but we shall, with the help of nullsets, define an equivalence relation in if, and it will turn out that every equivalence class contains representatives attaining values in R only. The linear operations will be defined for the equivalence classes via such finite-valued representatives. The space of equivalence classes will be a vector lattice in an obvious abstract sense.