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Non σ-Finite Measures and Product Measures
MATHEMATICS
NON o:-FINITE MEASURES AND PRODUCT MEASURES BY
N. G. DE BRUIJN (Communicated by Prof. H. D.
AND
A. C. ZAANEN
KLooSTERMAN
at the meeting of May 29, 1954)
l. Introduction. The present paper does not cont~;tin many new results; its principal aim is of a methodic (or, if one wants, didactic) nature. It shows how to build up the abstract theory of integration starting from one central notion, that of "measure", defined on as simple a collection of sets as possible, a "semiring" of sets, without the usual assumption of a-finiteness. A product measure of two not necessarily o:-finite measures is introduced, and the integral of a non-negative function is then defined as the product measure of its ordinate set, generalizing thus a well-known procedure in Euclidean space. This method of handling integration stands in a rather sharp contrast to the fashion of considering an integral first and foremost as a linear functional, originally defined on a simple class of functions, and which is to be extended then to a more extensive class. However, some remarks made in the last section of this paper may serve . perhaps as a hint to indicate that the contrast is not so great as sometimes believed or stated. We shall give a concise sketch of the theory. The theorems whos~ proofs are standard, are marked by an asterisk. Their proofs are omitted, and may be found e.g. in [l] or [2).
2. Measure on a semiring. We consider a general set X, the elements of which will be called points. All other point sets mentioned in what follows are subsets of X .. The empty set 0 is also considered to be a subset of X. We use standard notations: A-B for the set of all points belonging to A and not to B, E' for the complement of E (hence!E' =X -E), }.E, and liE, for union and intersection of an at most countable collection of sets E,, U.- E.- for the union of a possibly uncountable collection E .., A C B to indicate that A is a subset of B, and x E A to indicate that the point x belongs to the set A. Definition of a semiring. The collection r of point sets is called a semiring whenever (a) 0 E F. (b) If A E r, BE r, then ABE r. (c) If A E r, BE r, B c A, then A-B= }.C.. , where }.C.. is a finite or countable union, all c.. are disjoint and all c.. E r.
457
This definition is slightly more general than those in [1] and [2] since, in the present version, in condition (c) we also admit countable unions. Theorem l. If r is a semiring, and Al, ... , A,. E r, there exist disjoint sets Bv B 2 , ••• E r such that each A, is the union of a subsequence of the B;. Proof. If Sis a non-empty subset of the index set l, 2, ... , n, and S' the complementary subset, we define D8 =1Ikes.zes·(Ak-A 1), hence in particular D8 =1I~Ak if S' is empty. Then D 8 is a union of disjoint sets of and D 8 ,D8 , = 0 for S 1 =i=S2 • Since A,= L,8 D 8 , where the summation is over all S such that i E S, the proof is complete. Definition of a a-ring. A non-empty collection A of point sets is called a a-ring whenever (a) If A,. E A (n= l, 2, ... ), then L,;x>A,. EA. (b) If A EA, BEA, then A-BE A. * Theorem 2. If A is a a-ring, then (l) 0 EA. (2) If A,. E A (n= l, 2, ... ), then II;"' A,. E A, lim sup A,. E A and lim inf A,. E A. (3) A is a semiring.
r,
We shall adopt the usual conventions with regard to operations with ±=in the extended real number system; in particular 0·(± =)=0. Definition of measure on a semiring. Let there be assigned to every set A of a semiring r a real number ,u(A). This function ,u(A) is called a measure on whenever (a) ,u(O)=O and 0 ~ ,u(A) ~=for every A E r. (b) If A ET, A,. E T(n=l, 2, ... ),A C L,;x>A,., then ,u(A) ~ L,i"',u(A,.). (c) If A E r, A,. E r (n= l, ... , p) and disjoint, A J L,yA.,, then ,u(A) ~ L,y ,u(A .. ). * Theorem 3. If ,u(A) is a measure on the semiring r, then (l) ,u is monotone, that is, if A E T, BET, A C B, then ,u(A) ~ ,u(B). (2) ,u is countably additive, that is, if A E r, A .. E r (n= l, 2, ... ) and disjoint, A= !i"'A,., then ,u(A) = L,i"',u(A,.). These conditions, together with ,u(O) = 0, are also sufficient to ensure that ,u(A) is a measure on r.
r
We give some obvious examples, meant for future reference. ( l) X is real Euclidean space of m dimensions; T consists of 0 and all cells A (left open intervals ai
r
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of 0, X, all finite sets and their complements; ,u(O)=O, ,u(A)=n if A contains n points, ,u(A) = = in all other cases. (4) X is arbitrary, but not empty; P consists of all subsets of X; ,u(O)=O, ,u(A)== in all other cases. (5) X is arbitrary, but containing an uncountable number of points; r consists of all subsets of X; ,u(A)=O for countable A, ,u(A)= ex:> for uncountable A. (6) X is two-dimensional real Euclidean space (the plane therefore); r consists of 0 and all "horizontal" left open linesegments A(a
.u*(S) = inf !.u(A,) over all !A, "J S, all A,
is sequentially E
r.
If S fails to be sequentially covered, we define .u*(S) =ex:>. * Theorem I. We have (I) .u* is subadditive, that is, ,u*(!S,) ~ !.u*(S,). (2) .u*(O) = 0, and 0 :s;;; .u*(S) ~ ex:> for every S. (3) .u* is monotone, that is, if S C T, then .u*(S) ~ .u*(T). (4) If A E then .u*(A)=.u(A). * Theorem 2. (X, r, .u) and (X, F1 , .ut) generate the same exterior measure in X if and only if .u*=.u1 on F1 and .ui=.u on Comparison of the two variants of I, Ex (I) shows that .u*=.ut on F1 alone is not sufficient.
r,
r.
4.
Measurable sets.
The subadditivity of .u* shows that
.u*(S) :s;;; .u*(SE) + .u*(SE') for any pair of sets S, E. Definition of a measurable set. The set E C X measurable (or shortly measurable) whenever
~s
called .u-
.u*(S) = .u*(SE) + .u*(SE') for every S C X. The collection of all measurable sets is denoted by A. * Theorem I. If E E A, then E' EA. Hence 0 E A and X EA. * Theorem 2. If E., E A (n= I, 2, ... ),then !r'E, EA and lifE, EA.
459 If all En are disjoint, then p,*(SL,'f'En) = L,'f'p,*(SE,.) for any S. In particular p,*(L,'f'En)= L,f p,*(En), which shows that p,* is countably additive on A. Finally, if E 1 E A, E 2 E A, then E 1 -E2 EA. Hence A is a a-ring on which p,* is countably additive. * Theorem 3. If A E F, then A E A; hence FC A. It follows that p,* is a measure on 4, an extension of p, on Without fear for confusion we may and shall write, if E is measurable, p,(E) instead of p,*(E).
r.
If we consider A as a semiring, and start the process again, defining the exterior measure p,* by p,*(S) = inf '2,p,(E.. ) over all I,E .. ::> S, where all En E A, it follows from 3, Th. 2 that p,*==-p,*. The same is true if we start from some semiring F 1 satisfying r C F 1 C A. Theorem 4. E is measurable if and only if EA is measurable for each A E of finite measure. Proof. "Only if" is evident. Assume next that EA E A for each A E Fsatisfying p,(A) < oo. We have to prove that p,*(S) ~p,*(SE) + p,*(SE') for each S. This is true if p,*(S) = oo; assume therefore that p,*(S) < oo, so S C A= '2,'['An, where An E F, p,(An) < oo. Since all EAn E A by hypothesis, we have EA E A, which implies p,*(S)=p,*(SEA)+p,*{S(EA)'}. But SEA=SE, S(EA)'=S(E'+A')=SE', so that p,*(S)=fl*(SE)+p,*(SE'). Corollary. If p,(EA) = 0 for each A E of finite measure, then either p,(E) = 0 or p,(E) = oo. Proof. We immediately observe that p,(EF) = 0 for each measurable F of finite measure; hence, if p,(E) < oo, the choice F = E leads to p,(E) = 0.
r
r
vVe once more consider the examples of section 2. ( l) A is the collection of all Lebesgue measurable sets. The first variant gives the same collection A; for the second variant we have E E A if and only if E = '2,'['An, A,.. E F. (2) and (3) A is the collection of all subsets of X, p,(E) is the number of points in E. (4) and (5) A= F. (6) If E C X, denote for any fixed y by E 11 the set of all points (x, y) such that (x, y) E E. A is the collection of all sets U11 E 11 , where each E 11 is linearly Lebesgue measurable (cf. Theorem 4 above); p,(E) = oo if an uncountable number of the E 11 are non-empty, p,(E) = '2,11 p,(E11 ) if there is only a countable number of non-empty E 11 , where p,(E11 ) is the linear Lebesgue measure of E 11 • This example also shows that the following statements are both false: (a) If p,(EA)=O for each A E r, then p,(E)=O. (b) If p,(E F)= 0 for each F E A of finite measure, then p,(E) = 0. If Eisa set having exactly one point on each "horizontal" line, then E satisfies the hypotheses but not the conclusions of (a) and (b). 30
Series A
460
Properties of measurable sets 1. If p,*(E)=O, then E is measurable. Hence every subset of a set of measure zero is measurable and of measure zero. Corollary. If every Ae satisfies either p,(A)=O or p,(A)= oo, then every set E is measurable and satisfies either p,(E) = 0 or p,(E) = oo. Proof. Combine the above theorem with the corollary of 4, Th. 4. 5.
* Theorem
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We adopt the usual "almost everywhere" conventions. Any set 0= .!An, where all An E is called a a-set. If On(n= 1, 2, ... ) are a-sets, then .!~On and any finite intersection Ilf.On are a-sets. Any set IlfOn, where all On are a-sets, is called a a.,-set. The measurable set E is called of a-finite measure whenever E c .!An, where all An E and all p,(An) < oo. * Theorem 2. If Sis sequentially covered by then p,*(S)=inf p,(O) over all a-sets 0 covering S. For any set S we have p,*(S) = inf p,(E) over all measurable E coveringS (Take E=X whenever p,*(S)= oo). If E is of a-finite measure, there exists a a.,-set O.,=Il~On covering E such that p,(O.,-E)=O and the sequence on is descending. * Theorem 3. We have (a) If thesequenceEneA isascending,andE=limEn, thenp,(E)=limp,(En)· (b) If E,. E A is descending with limit E, and p,(E,.) < oo for some n, then p,(E) =lim p,(E,.). (c) If E,. e A, then I" (lim inf E*') ~lim inf p,(E,.). Furthermore, if p,(E,. + E,.+l + ... )< oo for some n, then p, (lim sup E,.) ~ lim sup p,(E,.); Hence, if E =lim E,. exiSts and p,(E,. + E,.+l + ... )< oo for some n, then p,(E) =lim p,(E,.).
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r
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6. Measurable functions. Let A be a a-ring of subsets of X. In this section the sets of A are called measurable sets, which is not meant to indicate that A is necessarily the a-ring of all measurable sets with respect to some measure p,. The function f(x), defined on the measurable set E, and assuming values in the extended real number system, is called measurable whenever E(f>a), i.e. the subset of E on which f(x)>a, is measurable for each finite a. The standard theorems on measurability of sums, products, inf, .sup, lim inf and lim sup of sequences of measurable functions are proved in the usual way. 7. Product measure. If X 1 and X 2 are point sets, and A C X 1 , B C X 2 , we denote by Ax B the Cartesian product of A and B (in particular AxO=OxB=O). If EC X 1 xX2 and x e X 1 , the set of ally e X 2 such that (x, y) EE is denoted by Ez. Similarly Ell for y Ex2. Theorem 1. If F 1 and F 2 are semirings in X 1 and X 2 respec.tively, then the collection of all subsets c =A X B of X= xl X x2 for which A E rl, B E is a semiring F= rl X r2. Proof. Trivial.
r2
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Theorem 2. If ,u1 (A) and ,u2(B) are measures on the semtnngs F 1 (in X 1 ) and F 2 (in X 2) respectively, and we define ,u(O) = f.ll(A),u 2(B) for each set 0 =A X B belonging to the semi ring = rl X r2 (we remind the reader that O·cx:>=cx:>·O=O), then ,u(O) is a measure on The proof is divided into several lemmas which together show that ,u has the desired properties. Lemma IX. We have ,u(O) = 0 and 0 ~ ,u(O) ~ ex:> for each 0 E If 0 1 E F, 0 2 E F and 0 1 C 0 2, then ,u(01 ) ~ ,u(02). Proof. Trivial. Lemma {3. If 0 E F, 0" E F (n,;, l, ... , p), all 0" are disjoint and 0 ::> 1YO,., then ,u(O) ~ 1Y,u(O.. ). Proof. Let O=A x B and O,.=A,. x B,. (n= l, ... , p). Then there exist disjoint sets Dv D 2, ... E F 1 such that each A,. is the union of a subsequence of the D,, and disjoint sets E 1 , E 2, ... E F 2 such that each B,. is the union of a subsequence of the E 1• If A,.='1D,, B,.='1E1, then A" X Bn='1'1D;, X E;, and ,u(Afl X B,.) = f.ll(A,.),u2(Bfl) = {'1,ul(D,)}{'1,u2(E;)}= '1'1,u(D, X E;).
r
r.
.
r.
Hence, observing that each D, x E 1 is contained in at most one of the A" x B", we find 1f,u(Ofl) ~ 1~11~1,u(D;,
X
E;)={1r'f.ll(D;,)}{!r'fl2(E;)}~,ul(A),u2(B)=,u(O).
This proof also shows that if 0 = 1¥0.. , then ,u(O) = 1f,u(O"). Lemma y. Let O, O, A C X 1 and ,ut(A)>a. Furthermore, let c xl X x2 and .u:( V.,) > b for almost every X EA. Then, if 0kEF(k=l,2, ... )and VC1fOk, we have 1r',u(Ok)>ab. Proof. We may assume that the Ok are disjoint (replace, if necessary, 10k by 0 1 +(02-01 )+ {(03 -01)-02}+ ... ).Now write Sn= 110k. For any .x E X 1 the measure ,u2 (S""') is an ascending function of n; hence, since .u:(V.,)>b almost everywhere on A and since V C 1r'Ok, we have ,u2 (S""')>b for n>n., almost everywhere on A. Writing Dn={xj,u2 (S""')>b}, the set D" is ,u1-measurable (since ,u2(S".,) assumes only a finite number of different values, each on a measurable set), the sequence D" is ascending, and almost every x E A belongs to some D... Hence, if D =lim D.. , we find on account of ,ut(A)>a that ,u1(D)>a. This shows that ,u1 (DN)>a for some suitable N. A subdivision (similarly as in the preceding lemma) leads now easily to 1f,u( Ok) > ab. Lemma b. If 0 E ok E (k= l, 2, ... ) and 0 c 1r'Ok, then ,u(O) ~ 1f,u(Ok). Proof. For ,u(O)=O there is nothing to prove; assume therefore that ,u(O) > 0. Then, ifO =A x B, there exist numbers a, bsuch that 0ab for all such pairs a, b. This is achieved by taking V = 0 in the preceding lemma.
v
r,
r
The measure ,u(O) obtained in this way on F= rl X r2 may now be extended to the a-ring A of all ,a-measurable sets in xl X x2. There is one
462 question which immediately arises: Assuming that F 1 and F 2 are not identical with the a-rings A1 and A 2 of all t-t1 -measurable and all f.t2measurable sets respectively, we can first extend f.ti and f.t2 from rl and r2 to A 1 and A 2 and then form, in the way sketched above, a measure fi on A 1 x A 2 • What is in this case the connection between f.t (on F 1 x F 2 ) and fi (on A 1 x A 2 )? Do they generate the same exterior measure (and therefore the same measurable sets) in X 1 x X 2 ? It is well-known that the answer to the last question is affirmative in case X 1 is of a-finite t-t1 -measure and X 2 of a-finite t-t2 -measure, but the same need not be true in the general case, as the following example shows: X 1 is the straight line, F 1 consists of 0 and all cells, t-t1 is Lebesgue measure; x2 is also the straight line, r2 consists of all subsets, f.t2(0) = 0 and t-t 2 (B) = oo for any non-empty B. We consider in X 1 x X 2 a set E = E 1 x E 2 consisting of one single point (x, y). Obviously fi(E)
=
t-t1 (E1 )t-t2 (E2 ) = 0 · oo = 0.
But the best we can do to cover E by sets A X B
E
rl X r2 is to take
A x B =A x E 2 where A is a cell, and t-t(A x B)= t-t1 (A )t-t2 ( B)= oo implies t-t*(E) = oo. Hence t-t*(E) i= fi(E). This failure is obviously due to the fact that E 2 , although consisting of only one point, is not of a-finite measure.
In order to avoid these difficulties we assume that T 1 = Av F 2 = A 2 , and that the product measure f.t = t-t1 x t-t2 on A 1 x A 2 is defined by t-t(El X E2) = f.t!(EI)f.t2(E2).
This measure f.t is then extended to the a-ring A of all t-t-measurable sets in X 1 x X 2 • If we consider Cartesian products of more than two point sets with measures defined on them, it is not possible to circumvent the stated difficulties, since the question of associativity arises, essentially the same question as the one under which conditions Fubini's theorem on repeated integrals holds. It seems that, in order to get a satisfactory result, it is necessary to assume that all point sets under consideration are of a-finite measure. Theorem 3. If E C X 1 x X 2 is measurable and of a-finite measure (it is not assumed that X 1 x X 2 itself is of a-finite measure), then E"' is t-t2 -measurable for almost every x E Xv that is, if Pis the set of all x for which E'" is not t-t2 -measurable, then t-t1 (P) == 0. If t-t(E) = 0, then t-t2 (E"') = 0 for almost every X E Xl. Proof. We first assume that t-t(E) = 0, and we denote, if b is such that Ob by A. Let us assume that t-t~(A)>O, so that O 0, in contradiction with t-t(E) = 0. Hence t-t1 (A) = 0. Denoting the set of all x such that t-t;(E'") > n-1 by A,., and the set of all x
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such that .uri(E.,) > 0 by P, we have now P= !An, hence ,u1 (P) = !.u1 (An) = 0. This completes the. proof in case ,u(E) = 0. Let now E =0 be an arbitrary a-set !An x Bn. Then, for every x, the set 0., is a countable union of sets Bn, and therefore ,u2-measurable. Finally, if E is an arbitrary ,u-measurable set of a-finite measure, there exists a descending sequence On (n= 1, 2, ... ) of a-sets covering E and a set B of measure zero such that E +S =liOn andES= 0. Hence, for every x, E.,+B.,=(IIOn)x=IIOna:, E'"S.,=O. Since all ona: are .U2-measurable for every X and s'" is .U2-measurable for almost every x, the set E., is ,u2-measurable for almost every x. If E is not of a-finite measure, the theorem is not necessarily true, not even in the case that one of the spaces X 1 or X 2 is of a-finite measure. We give two examples. Example 1. X 1 consists of one point, ,u1 (X1 )= =; X 2 is the straight line with ordinary Lebesgue measure. Every E c xl X x2 is measurable, since every 0 E A 1 x A 2 satisfies either ,u( 0) = 0 or ,u( 0) = = (cf. 5, Th. 1, Cor.). Take M C X 2 such that M is not ,u2-measurable. Then E=X1 x M is ,u-measurable, but "every" Ex, being equal to M, fails to be ,u2-measurable. The set of x for which E'" is not ,u2-measurable, is therefore of infinite measure. Example 2. (Xv Av ,u1 ) is. Example (6) in section 2 and section 4; X 2 is the straight line with ordinary Lebesgue measure. Since the points of X 1 are already denoted by (x, y), we denote the points of X 2 by z. We remind the reader that any set F C X 1 may be written as F=UyF11 , and that F is ,u1-measurable if and only if all Fy are ,u1-measurable. In the same way every E c xl X x2 may be written as E = uj/ Ey, and E is .u-measurable if and only if allEY are ,u-measurable. Let now S be a subset of X 1 such that every By fails to be .Ul-measurable, and let {IX} be the subset of x2 consisting of the single point z =IX. Consider now the subset U11 (8 11 x {y}) of xl X x2. Every Ey=Sy X {y} is .u-measurable (in fact, ,u(Ey)=O), so that E is ,u-measurable. Every E.=S. however fails to be ,u1 -measurable.
In Theorem 3 we derived that ,u(E) = 0 implies ,u2(E'") = 0 for almost every x. We shall now prove a result which goes in the converse direction. Theorem 4. Let E c xl X x2 be a .u-measurable set Stteh that ,u2 (E'") = 0 for almost every x E Xv and let A x B E A1 x A 2 . Then, if D=(A x B)-E, we have ,u(D)=,u(A x B). Hence, if E is of a-finite measure, ,u(E) = 0. Proof. If ,u(A x B)= 0, there is nothing to prove; assume therefore that ,u(A x B)=,u1 (A),u2(B)>0, and let 0
464 with f-l 1 (E 11 ) = 0 for ally, does not necessarily imply f-l(E) = 0, as the following example shows: X 1 is the straight line with f-l1 (A) = 0 for any countable A and f-l1 (A) = oo for any uncountable A ; f-l 2 is ordinary Lebesgue measure on the straight line x2. If E c xl Xx2 is the "diagonal" consisting of all points (x, x), then f-l 2(E.,) = 0 for all x E Xv f-l 1 (E11 ) = 0 for all y E X 2, and nevertheless f-l(E)= oo. 8.
The ordinate set of a non-negative function.
Let f-l be a measure in
X and m Lebesgue measure on the straight line R 1 . The product measure f-l x m in X x R 1 is denoted by jl.
Definition of ordinate set. If f(x);;?; 0 is defined on the setS C X, the ordinate set F of f(x) is the set of all (x, y) EX x R1 such that x E S and Oa) is f-l- measurable for each finite a, and by 4, Th. 4 it is sufficient for this purpose to prove that EaD is wmeasurable for each #-measurable DC X of finite measure. We may therefore restrict ourselves to the case that E is of finite #-measure and F of a-finite p-measure. For any y E R 1 the set F 11 is the set of all x E E such that f(x) >y> 0, hence F 11 =E(f(x)>y) for y > 0, and F 11 is empty for y ~ 0. Since on account of 7, Th. 3 the set F 11 is #-measurable for almost every y, this shows that E(f(x)>y) is #-measurable for all y=r,. in a suitably chosen sequence dense in R 1 . This however is sufficient to ensure the measurability of f(x). Conver~ely, if f(x) is #-measurable onE, and r,. is a dense sequence in R 1 , we define the set E (n=l, 2, ... )by E={x[x EE, f(x)>r,.} and the set B,. by B,.={y[O x B,. is p-measurable, and it will be sufficient to prove that F =S. If (x, y) E S, then (x, y) E E x Bk for some k, hence O x Bk, hence (x, y) ES. Sometimes the ordinate set of f(x) ;;?; 0 is defined as the set F* of all (x, y) such that O
465
(b) If F is measurable, then p,(F)=P,(F*), but p,(G)=O is not necessarily true. Proof. (a) Write H(tX) for the set of all points (x, tXy), where (x,y) eH. Evidently F*=li'm.r..oo H(1 + 1/n), F=li~00 H(1-1/n),
and so both F* and F are measurable. In order to prove the measurability of H 1 -F, we consider any set E C X x R 1 of finite p,-measure. Writing K = (H1 -F)E, we derive from 7, Th. 4 that p,(K) = 0, since each Kz consists of one point at most. Now 4, Th. 4 shows that H 1 -F is measurable. (b). If p,(G)>O, we have F-:JG( 1 / 2 )+G( 1 / 3 )+ ... , and so p,(F)=do. Therefore p,(F*)=P,(F). If p,(G)=O, then p,(F*)=P,(F) is trivial. If (cf. the example at the end of section 7) X is the straight line with p(E) = 0 for any countable E, and p(E) = oo for any uncountable E, and f(x)=x for x;;;::: 0, then p,(G)= oo. Nevertheless f(x) is measurable. In fact, any subset of X is measurable, and so any function defined on X is measurable. Theorem 3. If the measurable functions f.. (x) ;;;::: 0 (n= 1, 2, ... ) have the ordinate sets F .. , and h(x) =sup f..(x), k(x) =inf f..(x), p(x) =lim sup f,.(x), q(x)=lim inf f,.(x) have the ordinate sets H, K, P and Q respectively, then h(x}, k(x), p(x) and q(x) are measurable by section 6. Furthermore H =If F.,, and K, P, Q are contained in and of the same measure as Ilf F.,, lim sup F. and lim inf F" respectively. Proof. Wegivetheprooffor P. It is easily seen that PC lim supF,.CP*: Hence, in view of the preceeding theorem, p,(P.) = p, (lim sup F.,). 9. The integral. The introduction of the concept of an integral offers now no difficulties. As before, we assume that p is a measure in X, ·m Lebesgue measure on the straight line, and p, = p x m. If E C X is pmeasurable, and f(x) ;;;::: 0 is defined and p-measurable onE, the integral JBf(x)dp is defined to be the p,-measure of the ordinate set F of f(x). If fBfdp< oo we say that f(x) is integrable over E. Obviously, in this case, the subset of E on which f(x) > 0 is of a-finite measure. It is an immediate consequence of the countable additivity of p, that JDfdp is for DC E a countably additive set function on E, and that the monotone convergence theorem holds. The relation fB(f+g)dp=fBfdp+f~dp is first proved for g(x) a step function, and then by approximation for any measurable g(x);;;::: 0. Fatou's theorem follows by combining 5, Th. 3(c) and 8, Th. 3. Next, for a non-positive p-measurable f(x), we define fBfdp=- fB( -f)dp. Finally, for an arbitrary real p-measurable f(x}, we define
f+(x}=f(x) and j-(x)=O on E(f;;;?; 0}, f-(x)=f(x) and f+(x)=O on E(f
466
fEfdp,= fEf+dp,+ fEf-dp,. Whenever both f+(x) and f-(x) are integrable over E, the function f(x) is called integrable over E. The standard theorems
on integrable functions follow easily (the dominated convergence theorem for example by combining once more 5, Th. 3(c) and 8, Th. 3). Of course, one can also introduce the integral without using the product measure of the ordinate set. If E(f(x) =1= 0) is of a-finite measure one defines the integral first for step functions, and then by one of the standard methods for all measurable functions (cf. [1 ]). If f(x) ~ 0, and E(f > 0) is not of a-finite measure, one defines fEfdp, = oo. The theory of the Daniell integral fits naturally in the pattern which we have developed so far. There we have a vector space L of bounded real functions on a set X such that L is closed under the lattice operations f u g=max (f, g) and f n g=min (f, g). On L we assume to be defined a finite real functional!{!) satisfying l(iX.f+fJg)=od(f)+fJl(g) for real x, {J; l(f) ~ 0 for f(x) ~ 0; lim l(fn) = 0 if fn(x) is pointwise monotone decreasing to zero on X. Defining, for any f(x) ~ 0 belonging to L, the ordinate set F by F= {(x, y)lx EX, 0 ~ y
is pointwise monotone decreasing to zero, hence lim l(rk) = 0, which implies p,(A) = _Lfp,(An)· Finally, if A E r, An E r (n= 1, ... , p), all An are disjoint and A :l !fAn, then f(x)-g(x) ~ !f{fn(x)-gn(x)}, hence p,(A) ~ _Lfp,(An)·
Extending this measure fl to the collection A of all p,-measurable sets, and considering in particular those sets F E A which are ordinate sets of functions f(x) ~ 0, we extend the linear functional I to these f(x) by defining I (f)= p,(F). The extension to differences of such f(x) follows immediately, and the thus defined I (f) is called the Daniell integral of f(x) over X. If the original vector space L has the additional property that f E L implies f n 1 E L, it is possible in a well-known way to introduce a measure v in X such that I (f)= fxfdv for all f for which I (f) is defined .. University of Amsterdam Technical University of Delft REFERENCES 1. 2.
HALMOS, ZAANEN,
P. R., Measure Theory (New York; 1950). A. C., Linear Analysis (Amsterdam-Groningen, 1953).
NON o:-FINITE MEASURES AND PRODUCT MEASURES BY
N. G. DE BRUIJN (Communicated by Prof. H. D.
AND
A. C. ZAANEN
KLooSTERMAN
at the meeting of May 29, 1954)
l. Introduction. The present paper does not cont~;tin many new results; its principal aim is of a methodic (or, if one wants, didactic) nature. It shows how to build up the abstract theory of integration starting from one central notion, that of "measure", defined on as simple a collection of sets as possible, a "semiring" of sets, without the usual assumption of a-finiteness. A product measure of two not necessarily o:-finite measures is introduced, and the integral of a non-negative function is then defined as the product measure of its ordinate set, generalizing thus a well-known procedure in Euclidean space. This method of handling integration stands in a rather sharp contrast to the fashion of considering an integral first and foremost as a linear functional, originally defined on a simple class of functions, and which is to be extended then to a more extensive class. However, some remarks made in the last section of this paper may serve . perhaps as a hint to indicate that the contrast is not so great as sometimes believed or stated. We shall give a concise sketch of the theory. The theorems whos~ proofs are standard, are marked by an asterisk. Their proofs are omitted, and may be found e.g. in [l] or [2).
2. Measure on a semiring. We consider a general set X, the elements of which will be called points. All other point sets mentioned in what follows are subsets of X .. The empty set 0 is also considered to be a subset of X. We use standard notations: A-B for the set of all points belonging to A and not to B, E' for the complement of E (hence!E' =X -E), }.E, and liE, for union and intersection of an at most countable collection of sets E,, U.- E.- for the union of a possibly uncountable collection E .., A C B to indicate that A is a subset of B, and x E A to indicate that the point x belongs to the set A. Definition of a semiring. The collection r of point sets is called a semiring whenever (a) 0 E F. (b) If A E r, BE r, then ABE r. (c) If A E r, BE r, B c A, then A-B= }.C.. , where }.C.. is a finite or countable union, all c.. are disjoint and all c.. E r.
457
This definition is slightly more general than those in [1] and [2] since, in the present version, in condition (c) we also admit countable unions. Theorem l. If r is a semiring, and Al, ... , A,. E r, there exist disjoint sets Bv B 2 , ••• E r such that each A, is the union of a subsequence of the B;. Proof. If Sis a non-empty subset of the index set l, 2, ... , n, and S' the complementary subset, we define D8 =1Ikes.zes·(Ak-A 1), hence in particular D8 =1I~Ak if S' is empty. Then D 8 is a union of disjoint sets of and D 8 ,D8 , = 0 for S 1 =i=S2 • Since A,= L,8 D 8 , where the summation is over all S such that i E S, the proof is complete. Definition of a a-ring. A non-empty collection A of point sets is called a a-ring whenever (a) If A,. E A (n= l, 2, ... ), then L,;x>A,. EA. (b) If A EA, BEA, then A-BE A. * Theorem 2. If A is a a-ring, then (l) 0 EA. (2) If A,. E A (n= l, 2, ... ), then II;"' A,. E A, lim sup A,. E A and lim inf A,. E A. (3) A is a semiring.
r,
We shall adopt the usual conventions with regard to operations with ±=in the extended real number system; in particular 0·(± =)=0. Definition of measure on a semiring. Let there be assigned to every set A of a semiring r a real number ,u(A). This function ,u(A) is called a measure on whenever (a) ,u(O)=O and 0 ~ ,u(A) ~=for every A E r. (b) If A ET, A,. E T(n=l, 2, ... ),A C L,;x>A,., then ,u(A) ~ L,i"',u(A,.). (c) If A E r, A,. E r (n= l, ... , p) and disjoint, A J L,yA.,, then ,u(A) ~ L,y ,u(A .. ). * Theorem 3. If ,u(A) is a measure on the semiring r, then (l) ,u is monotone, that is, if A E T, BET, A C B, then ,u(A) ~ ,u(B). (2) ,u is countably additive, that is, if A E r, A .. E r (n= l, 2, ... ) and disjoint, A= !i"'A,., then ,u(A) = L,i"',u(A,.). These conditions, together with ,u(O) = 0, are also sufficient to ensure that ,u(A) is a measure on r.
r
We give some obvious examples, meant for future reference. ( l) X is real Euclidean space of m dimensions; T consists of 0 and all cells A (left open intervals ai
r
458
of 0, X, all finite sets and their complements; ,u(O)=O, ,u(A)=n if A contains n points, ,u(A) = = in all other cases. (4) X is arbitrary, but not empty; P consists of all subsets of X; ,u(O)=O, ,u(A)== in all other cases. (5) X is arbitrary, but containing an uncountable number of points; r consists of all subsets of X; ,u(A)=O for countable A, ,u(A)= ex:> for uncountable A. (6) X is two-dimensional real Euclidean space (the plane therefore); r consists of 0 and all "horizontal" left open linesegments A(a
.u*(S) = inf !.u(A,) over all !A, "J S, all A,
is sequentially E
r.
If S fails to be sequentially covered, we define .u*(S) =ex:>. * Theorem I. We have (I) .u* is subadditive, that is, ,u*(!S,) ~ !.u*(S,). (2) .u*(O) = 0, and 0 :s;;; .u*(S) ~ ex:> for every S. (3) .u* is monotone, that is, if S C T, then .u*(S) ~ .u*(T). (4) If A E then .u*(A)=.u(A). * Theorem 2. (X, r, .u) and (X, F1 , .ut) generate the same exterior measure in X if and only if .u*=.u1 on F1 and .ui=.u on Comparison of the two variants of I, Ex (I) shows that .u*=.ut on F1 alone is not sufficient.
r,
r.
4.
Measurable sets.
The subadditivity of .u* shows that
.u*(S) :s;;; .u*(SE) + .u*(SE') for any pair of sets S, E. Definition of a measurable set. The set E C X measurable (or shortly measurable) whenever
~s
called .u-
.u*(S) = .u*(SE) + .u*(SE') for every S C X. The collection of all measurable sets is denoted by A. * Theorem I. If E E A, then E' EA. Hence 0 E A and X EA. * Theorem 2. If E., E A (n= I, 2, ... ),then !r'E, EA and lifE, EA.
459 If all En are disjoint, then p,*(SL,'f'En) = L,'f'p,*(SE,.) for any S. In particular p,*(L,'f'En)= L,f p,*(En), which shows that p,* is countably additive on A. Finally, if E 1 E A, E 2 E A, then E 1 -E2 EA. Hence A is a a-ring on which p,* is countably additive. * Theorem 3. If A E F, then A E A; hence FC A. It follows that p,* is a measure on 4, an extension of p, on Without fear for confusion we may and shall write, if E is measurable, p,(E) instead of p,*(E).
r.
If we consider A as a semiring, and start the process again, defining the exterior measure p,* by p,*(S) = inf '2,p,(E.. ) over all I,E .. ::> S, where all En E A, it follows from 3, Th. 2 that p,*==-p,*. The same is true if we start from some semiring F 1 satisfying r C F 1 C A. Theorem 4. E is measurable if and only if EA is measurable for each A E of finite measure. Proof. "Only if" is evident. Assume next that EA E A for each A E Fsatisfying p,(A) < oo. We have to prove that p,*(S) ~p,*(SE) + p,*(SE') for each S. This is true if p,*(S) = oo; assume therefore that p,*(S) < oo, so S C A= '2,'['An, where An E F, p,(An) < oo. Since all EAn E A by hypothesis, we have EA E A, which implies p,*(S)=p,*(SEA)+p,*{S(EA)'}. But SEA=SE, S(EA)'=S(E'+A')=SE', so that p,*(S)=fl*(SE)+p,*(SE'). Corollary. If p,(EA) = 0 for each A E of finite measure, then either p,(E) = 0 or p,(E) = oo. Proof. We immediately observe that p,(EF) = 0 for each measurable F of finite measure; hence, if p,(E) < oo, the choice F = E leads to p,(E) = 0.
r
r
vVe once more consider the examples of section 2. ( l) A is the collection of all Lebesgue measurable sets. The first variant gives the same collection A; for the second variant we have E E A if and only if E = '2,'['An, A,.. E F. (2) and (3) A is the collection of all subsets of X, p,(E) is the number of points in E. (4) and (5) A= F. (6) If E C X, denote for any fixed y by E 11 the set of all points (x, y) such that (x, y) E E. A is the collection of all sets U11 E 11 , where each E 11 is linearly Lebesgue measurable (cf. Theorem 4 above); p,(E) = oo if an uncountable number of the E 11 are non-empty, p,(E) = '2,11 p,(E11 ) if there is only a countable number of non-empty E 11 , where p,(E11 ) is the linear Lebesgue measure of E 11 • This example also shows that the following statements are both false: (a) If p,(EA)=O for each A E r, then p,(E)=O. (b) If p,(E F)= 0 for each F E A of finite measure, then p,(E) = 0. If Eisa set having exactly one point on each "horizontal" line, then E satisfies the hypotheses but not the conclusions of (a) and (b). 30
Series A
460
Properties of measurable sets 1. If p,*(E)=O, then E is measurable. Hence every subset of a set of measure zero is measurable and of measure zero. Corollary. If every Ae satisfies either p,(A)=O or p,(A)= oo, then every set E is measurable and satisfies either p,(E) = 0 or p,(E) = oo. Proof. Combine the above theorem with the corollary of 4, Th. 4. 5.
* Theorem
r
We adopt the usual "almost everywhere" conventions. Any set 0= .!An, where all An E is called a a-set. If On(n= 1, 2, ... ) are a-sets, then .!~On and any finite intersection Ilf.On are a-sets. Any set IlfOn, where all On are a-sets, is called a a.,-set. The measurable set E is called of a-finite measure whenever E c .!An, where all An E and all p,(An) < oo. * Theorem 2. If Sis sequentially covered by then p,*(S)=inf p,(O) over all a-sets 0 covering S. For any set S we have p,*(S) = inf p,(E) over all measurable E coveringS (Take E=X whenever p,*(S)= oo). If E is of a-finite measure, there exists a a.,-set O.,=Il~On covering E such that p,(O.,-E)=O and the sequence on is descending. * Theorem 3. We have (a) If thesequenceEneA isascending,andE=limEn, thenp,(E)=limp,(En)· (b) If E,. E A is descending with limit E, and p,(E,.) < oo for some n, then p,(E) =lim p,(E,.). (c) If E,. e A, then I" (lim inf E*') ~lim inf p,(E,.). Furthermore, if p,(E,. + E,.+l + ... )< oo for some n, then p, (lim sup E,.) ~ lim sup p,(E,.); Hence, if E =lim E,. exiSts and p,(E,. + E,.+l + ... )< oo for some n, then p,(E) =lim p,(E,.).
r,
r
r,
6. Measurable functions. Let A be a a-ring of subsets of X. In this section the sets of A are called measurable sets, which is not meant to indicate that A is necessarily the a-ring of all measurable sets with respect to some measure p,. The function f(x), defined on the measurable set E, and assuming values in the extended real number system, is called measurable whenever E(f>a), i.e. the subset of E on which f(x)>a, is measurable for each finite a. The standard theorems on measurability of sums, products, inf, .sup, lim inf and lim sup of sequences of measurable functions are proved in the usual way. 7. Product measure. If X 1 and X 2 are point sets, and A C X 1 , B C X 2 , we denote by Ax B the Cartesian product of A and B (in particular AxO=OxB=O). If EC X 1 xX2 and x e X 1 , the set of ally e X 2 such that (x, y) EE is denoted by Ez. Similarly Ell for y Ex2. Theorem 1. If F 1 and F 2 are semirings in X 1 and X 2 respec.tively, then the collection of all subsets c =A X B of X= xl X x2 for which A E rl, B E is a semiring F= rl X r2. Proof. Trivial.
r2
461
Theorem 2. If ,u1 (A) and ,u2(B) are measures on the semtnngs F 1 (in X 1 ) and F 2 (in X 2) respectively, and we define ,u(O) = f.ll(A),u 2(B) for each set 0 =A X B belonging to the semi ring = rl X r2 (we remind the reader that O·cx:>=cx:>·O=O), then ,u(O) is a measure on The proof is divided into several lemmas which together show that ,u has the desired properties. Lemma IX. We have ,u(O) = 0 and 0 ~ ,u(O) ~ ex:> for each 0 E If 0 1 E F, 0 2 E F and 0 1 C 0 2, then ,u(01 ) ~ ,u(02). Proof. Trivial. Lemma {3. If 0 E F, 0" E F (n,;, l, ... , p), all 0" are disjoint and 0 ::> 1YO,., then ,u(O) ~ 1Y,u(O.. ). Proof. Let O=A x B and O,.=A,. x B,. (n= l, ... , p). Then there exist disjoint sets Dv D 2, ... E F 1 such that each A,. is the union of a subsequence of the D,, and disjoint sets E 1 , E 2, ... E F 2 such that each B,. is the union of a subsequence of the E 1• If A,.='1D,, B,.='1E1, then A" X Bn='1'1D;, X E;, and ,u(Afl X B,.) = f.ll(A,.),u2(Bfl) = {'1,ul(D,)}{'1,u2(E;)}= '1'1,u(D, X E;).
r
r.
.
r.
Hence, observing that each D, x E 1 is contained in at most one of the A" x B", we find 1f,u(Ofl) ~ 1~11~1,u(D;,
X
E;)={1r'f.ll(D;,)}{!r'fl2(E;)}~,ul(A),u2(B)=,u(O).
This proof also shows that if 0 = 1¥0.. , then ,u(O) = 1f,u(O"). Lemma y. Let O
v
r,
r
The measure ,u(O) obtained in this way on F= rl X r2 may now be extended to the a-ring A of all ,a-measurable sets in xl X x2. There is one
462 question which immediately arises: Assuming that F 1 and F 2 are not identical with the a-rings A1 and A 2 of all t-t1 -measurable and all f.t2measurable sets respectively, we can first extend f.ti and f.t2 from rl and r2 to A 1 and A 2 and then form, in the way sketched above, a measure fi on A 1 x A 2 • What is in this case the connection between f.t (on F 1 x F 2 ) and fi (on A 1 x A 2 )? Do they generate the same exterior measure (and therefore the same measurable sets) in X 1 x X 2 ? It is well-known that the answer to the last question is affirmative in case X 1 is of a-finite t-t1 -measure and X 2 of a-finite t-t2 -measure, but the same need not be true in the general case, as the following example shows: X 1 is the straight line, F 1 consists of 0 and all cells, t-t1 is Lebesgue measure; x2 is also the straight line, r2 consists of all subsets, f.t2(0) = 0 and t-t 2 (B) = oo for any non-empty B. We consider in X 1 x X 2 a set E = E 1 x E 2 consisting of one single point (x, y). Obviously fi(E)
=
t-t1 (E1 )t-t2 (E2 ) = 0 · oo = 0.
But the best we can do to cover E by sets A X B
E
rl X r2 is to take
A x B =A x E 2 where A is a cell, and t-t(A x B)= t-t1 (A )t-t2 ( B)= oo implies t-t*(E) = oo. Hence t-t*(E) i= fi(E). This failure is obviously due to the fact that E 2 , although consisting of only one point, is not of a-finite measure.
In order to avoid these difficulties we assume that T 1 = Av F 2 = A 2 , and that the product measure f.t = t-t1 x t-t2 on A 1 x A 2 is defined by t-t(El X E2) = f.t!(EI)f.t2(E2).
This measure f.t is then extended to the a-ring A of all t-t-measurable sets in X 1 x X 2 • If we consider Cartesian products of more than two point sets with measures defined on them, it is not possible to circumvent the stated difficulties, since the question of associativity arises, essentially the same question as the one under which conditions Fubini's theorem on repeated integrals holds. It seems that, in order to get a satisfactory result, it is necessary to assume that all point sets under consideration are of a-finite measure. Theorem 3. If E C X 1 x X 2 is measurable and of a-finite measure (it is not assumed that X 1 x X 2 itself is of a-finite measure), then E"' is t-t2 -measurable for almost every x E Xv that is, if Pis the set of all x for which E'" is not t-t2 -measurable, then t-t1 (P) == 0. If t-t(E) = 0, then t-t2 (E"') = 0 for almost every X E Xl. Proof. We first assume that t-t(E) = 0, and we denote, if b is such that O
463
such that .uri(E.,) > 0 by P, we have now P= !An, hence ,u1 (P) = !.u1 (An) = 0. This completes the. proof in case ,u(E) = 0. Let now E =0 be an arbitrary a-set !An x Bn. Then, for every x, the set 0., is a countable union of sets Bn, and therefore ,u2-measurable. Finally, if E is an arbitrary ,u-measurable set of a-finite measure, there exists a descending sequence On (n= 1, 2, ... ) of a-sets covering E and a set B of measure zero such that E +S =liOn andES= 0. Hence, for every x, E.,+B.,=(IIOn)x=IIOna:, E'"S.,=O. Since all ona: are .U2-measurable for every X and s'" is .U2-measurable for almost every x, the set E., is ,u2-measurable for almost every x. If E is not of a-finite measure, the theorem is not necessarily true, not even in the case that one of the spaces X 1 or X 2 is of a-finite measure. We give two examples. Example 1. X 1 consists of one point, ,u1 (X1 )= =; X 2 is the straight line with ordinary Lebesgue measure. Every E c xl X x2 is measurable, since every 0 E A 1 x A 2 satisfies either ,u( 0) = 0 or ,u( 0) = = (cf. 5, Th. 1, Cor.). Take M C X 2 such that M is not ,u2-measurable. Then E=X1 x M is ,u-measurable, but "every" Ex, being equal to M, fails to be ,u2-measurable. The set of x for which E'" is not ,u2-measurable, is therefore of infinite measure. Example 2. (Xv Av ,u1 ) is. Example (6) in section 2 and section 4; X 2 is the straight line with ordinary Lebesgue measure. Since the points of X 1 are already denoted by (x, y), we denote the points of X 2 by z. We remind the reader that any set F C X 1 may be written as F=UyF11 , and that F is ,u1-measurable if and only if all Fy are ,u1-measurable. In the same way every E c xl X x2 may be written as E = uj/ Ey, and E is .u-measurable if and only if allEY are ,u-measurable. Let now S be a subset of X 1 such that every By fails to be .Ul-measurable, and let {IX} be the subset of x2 consisting of the single point z =IX. Consider now the subset U11 (8 11 x {y}) of xl X x2. Every Ey=Sy X {y} is .u-measurable (in fact, ,u(Ey)=O), so that E is ,u-measurable. Every E.=S. however fails to be ,u1 -measurable.
In Theorem 3 we derived that ,u(E) = 0 implies ,u2(E'") = 0 for almost every x. We shall now prove a result which goes in the converse direction. Theorem 4. Let E c xl X x2 be a .u-measurable set Stteh that ,u2 (E'") = 0 for almost every x E Xv and let A x B E A1 x A 2 . Then, if D=(A x B)-E, we have ,u(D)=,u(A x B). Hence, if E is of a-finite measure, ,u(E) = 0. Proof. If ,u(A x B)= 0, there is nothing to prove; assume therefore that ,u(A x B)=,u1 (A),u2(B)>0, and let 0
464 with f-l 1 (E 11 ) = 0 for ally, does not necessarily imply f-l(E) = 0, as the following example shows: X 1 is the straight line with f-l1 (A) = 0 for any countable A and f-l1 (A) = oo for any uncountable A ; f-l 2 is ordinary Lebesgue measure on the straight line x2. If E c xl Xx2 is the "diagonal" consisting of all points (x, x), then f-l 2(E.,) = 0 for all x E Xv f-l 1 (E11 ) = 0 for all y E X 2, and nevertheless f-l(E)= oo. 8.
The ordinate set of a non-negative function.
Let f-l be a measure in
X and m Lebesgue measure on the straight line R 1 . The product measure f-l x m in X x R 1 is denoted by jl.
Definition of ordinate set. If f(x);;?; 0 is defined on the setS C X, the ordinate set F of f(x) is the set of all (x, y) EX x R1 such that x E S and O
465
(b) If F is measurable, then p,(F)=P,(F*), but p,(G)=O is not necessarily true. Proof. (a) Write H(tX) for the set of all points (x, tXy), where (x,y) eH. Evidently F*=li'm.r..oo H(1 + 1/n), F=li~00 H(1-1/n),
and so both F* and F are measurable. In order to prove the measurability of H 1 -F, we consider any set E C X x R 1 of finite p,-measure. Writing K = (H1 -F)E, we derive from 7, Th. 4 that p,(K) = 0, since each Kz consists of one point at most. Now 4, Th. 4 shows that H 1 -F is measurable. (b). If p,(G)>O, we have F-:JG( 1 / 2 )+G( 1 / 3 )+ ... , and so p,(F)=do. Therefore p,(F*)=P,(F). If p,(G)=O, then p,(F*)=P,(F) is trivial. If (cf. the example at the end of section 7) X is the straight line with p(E) = 0 for any countable E, and p(E) = oo for any uncountable E, and f(x)=x for x;;;::: 0, then p,(G)= oo. Nevertheless f(x) is measurable. In fact, any subset of X is measurable, and so any function defined on X is measurable. Theorem 3. If the measurable functions f.. (x) ;;;::: 0 (n= 1, 2, ... ) have the ordinate sets F .. , and h(x) =sup f..(x), k(x) =inf f..(x), p(x) =lim sup f,.(x), q(x)=lim inf f,.(x) have the ordinate sets H, K, P and Q respectively, then h(x}, k(x), p(x) and q(x) are measurable by section 6. Furthermore H =If F.,, and K, P, Q are contained in and of the same measure as Ilf F.,, lim sup F. and lim inf F" respectively. Proof. Wegivetheprooffor P. It is easily seen that PC lim supF,.CP*: Hence, in view of the preceeding theorem, p,(P.) = p, (lim sup F.,). 9. The integral. The introduction of the concept of an integral offers now no difficulties. As before, we assume that p is a measure in X, ·m Lebesgue measure on the straight line, and p, = p x m. If E C X is pmeasurable, and f(x) ;;;::: 0 is defined and p-measurable onE, the integral JBf(x)dp is defined to be the p,-measure of the ordinate set F of f(x). If fBfdp< oo we say that f(x) is integrable over E. Obviously, in this case, the subset of E on which f(x) > 0 is of a-finite measure. It is an immediate consequence of the countable additivity of p, that JDfdp is for DC E a countably additive set function on E, and that the monotone convergence theorem holds. The relation fB(f+g)dp=fBfdp+f~dp is first proved for g(x) a step function, and then by approximation for any measurable g(x);;;::: 0. Fatou's theorem follows by combining 5, Th. 3(c) and 8, Th. 3. Next, for a non-positive p-measurable f(x), we define fBfdp=- fB( -f)dp. Finally, for an arbitrary real p-measurable f(x}, we define
f+(x}=f(x) and j-(x)=O on E(f;;;?; 0}, f-(x)=f(x) and f+(x)=O on E(f
466
fEfdp,= fEf+dp,+ fEf-dp,. Whenever both f+(x) and f-(x) are integrable over E, the function f(x) is called integrable over E. The standard theorems
on integrable functions follow easily (the dominated convergence theorem for example by combining once more 5, Th. 3(c) and 8, Th. 3). Of course, one can also introduce the integral without using the product measure of the ordinate set. If E(f(x) =1= 0) is of a-finite measure one defines the integral first for step functions, and then by one of the standard methods for all measurable functions (cf. [1 ]). If f(x) ~ 0, and E(f > 0) is not of a-finite measure, one defines fEfdp, = oo. The theory of the Daniell integral fits naturally in the pattern which we have developed so far. There we have a vector space L of bounded real functions on a set X such that L is closed under the lattice operations f u g=max (f, g) and f n g=min (f, g). On L we assume to be defined a finite real functional!{!) satisfying l(iX.f+fJg)=od(f)+fJl(g) for real x, {J; l(f) ~ 0 for f(x) ~ 0; lim l(fn) = 0 if fn(x) is pointwise monotone decreasing to zero on X. Defining, for any f(x) ~ 0 belonging to L, the ordinate set F by F= {(x, y)lx EX, 0 ~ y
is pointwise monotone decreasing to zero, hence lim l(rk) = 0, which implies p,(A) = _Lfp,(An)· Finally, if A E r, An E r (n= 1, ... , p), all An are disjoint and A :l !fAn, then f(x)-g(x) ~ !f{fn(x)-gn(x)}, hence p,(A) ~ _Lfp,(An)·
Extending this measure fl to the collection A of all p,-measurable sets, and considering in particular those sets F E A which are ordinate sets of functions f(x) ~ 0, we extend the linear functional I to these f(x) by defining I (f)= p,(F). The extension to differences of such f(x) follows immediately, and the thus defined I (f) is called the Daniell integral of f(x) over X. If the original vector space L has the additional property that f E L implies f n 1 E L, it is possible in a well-known way to introduce a measure v in X such that I (f)= fxfdv for all f for which I (f) is defined .. University of Amsterdam Technical University of Delft REFERENCES 1. 2.
HALMOS, ZAANEN,
P. R., Measure Theory (New York; 1950). A. C., Linear Analysis (Amsterdam-Groningen, 1953).