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8.8: Product Measures. Iterated Integrals

( \newcommand{\kernel}{\mathrm{null}\,}\)

Let (X,M,m) and (Y,N,n) be measure spaces, with XM and YN. Let C be the family of all "rectangles," i.e., sets

A×B,

with AM,BN,mA<, and nB<.

Define a premeasure s:CE1 by

s(A×B)=mAnB,A×BC.

Let p be the s-induced outer measure in X×Y and

p:PE

the p-induced measure ("product measure," p=m×n) on the σ-field P of all p-measurable sets in X×Y (Chapter 7, §§5-6).

We consider functions f:X×YE (extended-real).

I. We begin with some definitions.

Definitions

(1) Given a function f:XYE (of two variables x,y), let fx or f(x,) denote the function on Y given by

fx(y)=f(x,y);

it arises from f by fixing x.

Similarly, fy or f(,y) is given by fy(x)=f(x,y).

(2) Define g:XE by

g(x)=Yf(x,)dn,

and set

XYfdndm=Xgdm,

also written

Xdm(x)Yf(x,y)dn(y).

This is called the iterated integral of f on Y and X, in this order.

Similarly,

h(y)=Xfydm

and

YXfdmdn=Yhdn.

Note that by the rules of §5, these integrals are always defined.

(3) With f,g,h as above, we say that f is a Fubini map or has the Fubini properties (after the mathematician Fubini) iff

(a) g is m-measurable on X and h is n-measurable on Y;

(b) fx is n-measurable on Y for almost all x (i.e., for xXQ, mQ=0);fy is m-measurable on X for yYQ,nQ=0; and

(c) the iterated integrals above satisfy

XYfdndm=YXfdmdn=X×Yfdp

(the main point).

For monotone sequences

fk:X×YE(k=1,2,),

we now obtain the following lemma.

Lemma 8.8.1

If 0fkf (pointwise) on X×Y and if each fk has Fubini property (a), (b), or (c), then f has the same property.

Proof

For k=1,2,, set

gk(x)=Yfk(x,)dn

and

hk(y)=Xfk(,y)dm.

By assumpsion,

0fk(x,)f(x,)

pointwise on Y. Thus by Theorem 4 in §6,

Yfk(x,)Yf(x,)dn,

i.e., gkg (pointwise) on X, with g as in Definition 2.

Again, by Theorem 4 of §6,

XgkdmXgdm;

or by Definition 2,

XYfdndm=limkXYfkdndm.

Similarly for

YXfdmdn

and

X×Yfdp.

Hence f satisfies (c) if all fk do.

Next, let fk have property (b); so (k)fk(x,) is n-measurable on Y if xXQk (mQk=0). Let

Q=k=1Qk;

so mQ=0, and all fk(x,) are n-measurable on Y, for xXQ. Hence so is

f(x,)=limkfk(x,).

Similarly for f(,y). Thus f satisfies (b).

Property (a) follows from gkg and hkh.

Using Problems 9 and 10 from §6, the reader will also easily verify the following lemma.

Lemma 8.8.2

(i) If f1 and f2 are nonnegative, p-measurable Fubini maps, so is af1+bf2 for a,b0.

(ii) If, in addition,

X×Yf1dp< or X×Yf2dp<,

then f1f2 is a Fubini map, too

Lemma 8.8.3

Let f=i=1fi (pointwise), with fi0 on X×Y.

(i) If all fi are p-measurable Fubini maps, so is f.

(ii) If the fi have Fubini properties (a) and (b), then

XYfdndm=i=1XYfidndm

and

YXfdmdn=i=1YXfidmdn.

II. By Theorem 4 of Chapter 7, §3, the family C (see above) is a semiring, being the product of two rings,

{AM|mA<} and {BN|nB<}.

(Verify!) Thus using Theorem 2 in Chapter 7, §6, we now show that p is an extension of s:CE1.

Theorem 8.8.1

The product premeasure s is σ-additive on the semiring C. Hence

(i) CP and p=s< on C, and

(ii) the characteristic function CD of any set DC is a Fubini map.

Proof

Let D=A×BC; so

CD(x,y)=CA(x)CB(y).

(Why?) Thus for a fixed x,CD(x,) is just a multiple of the N-simple map CB, hence n-measurable on Y. Also,

g(x)=YCD(x,)dn=CA(x)YCBdn=CA(x)nB;

so g=CAnB is M-simple on X, with

XYCDdndm=Xgdm=nBXCAdm=nBmA=sD.

Similarly for CD(,y), and

h(y)=XCD(,y)dm.

Thus CD has Fubini properties (a) and (b), and for every DC

XYCDdndm=YXCDdmdn=sD.

To prove σ-additivity, let

D=i=1Di (disjoint), DiC;

so

CD=i=1CDi.

(Why?) As shown above, each CDi has Fubini properties (a) and (b); so by (1) and Lemma 3,

sD=XYCDdndm=i=1XYCDidndm=i=1sDi,

as required.

Assertion (i) now follows by Theorem 2 in Chapter 7, §6. Hence

sD=pD=X×YCDdp;

so by formula (1), CD also has Fubini property (c), and all is proved.

Next, let P be the σ-ring generated by the semiring C (so CPP).

Lemma 8.8.4

P is the least set family R such that

(i) RC;

(ii) R is closed under countable disjoint unions; and

(iii) HDR if DR and DH,HC.

This is simply Theorem 3 in Chapter 7, §3, with changed notation.

Lemma 8.8.5

If DP (σ-generated by C), then CD is a Fubini map.

Proof

Let R be the family of all DP such that CD is a Fubini map. We shall show that R satisfies (i)-(iii) of Lemma 4, and so PR.

(ii) Let

D=i=1Di (disjoint),DiR.

Then

CD=i=1CDi,

and each CDi is a Fubini map. Hence so is CD by Lemma 3. Thus DR, proving (ii).

(iii) We must show that CHD is a Fubini map if CD is and if DH,HC. Now, DH implies

CHD=CHCD.

(Why?) Also, by Theorem 1, HC implies

X×YCHdp=pH=sH<,

and CH is a Fubini map. So is CD by assumption. So also is

CHD=CHCD

by Lemma 2(ii). Thus HDR, proving (iii).

By Lemma 4, then, PR. Hence (DP)CD is a Fubini map.

We can now establish one of the main theorems, due to Fubini.

Theorem 8.8.2 (Fubini I)

Suppose f:X×YE is P-measurable on X×Y (P as above) rom. Then f is a Fubini map if either

(i) f0 on X×Y, or

(ii) one of

X×Y|f|dp,XY|f|dndm,orYX|f|dmdn

is finite.

In both cases,

XYfdndm=YXfdmdn=X×Yfdp.

Proof

First, let

f=i=1aiCDi(ai0,DiP),

i.e., f is P-elementary, hence certainly p-measurable. (Why?) By Lemmas 5 and 2, each aiCDi is a Fubini map. Hence so is f (Lemma 3). Formula (2) is simply Fubini property (c).

Now take any P-measurable f0. By Lemma 2 in §2,

f=limkfk on X×Y

for some sequence {fk} of P-elementary maps, fk0. As shown above, each fk is a Fubini map. Hence so is f by Lemma 1. This settles case (i).

Next, assume (ii). As f is P-measurable, so are f+,f, and |f| (Theorem 2 in §2). As they are nonnegative, they are Fubini maps by case (i).

So is f=f+f by Lemma 2(ii), since f+|f| implies

X×Yf+dp<

by our assumption (ii). (The three integrals are equal, as |f| is a Fubini map.)

Thus all is proved.

III. We now want to replace P by P in Lemma 5 and Theorem 2. This works only under certain σ-finiteness conditions, as shown below.

Lemma 8.8.6

Let DP be σ-finite, i.e.,

D=i=1Di (disjoint)

for some DiP, with pDi< (i=1,2,).

Then there is aKP such that p(KD)=0 and DK.

Proof

As P is a σ-ring containing C, it also contains Cσ. Thus by Theorem 3 of Chapter 7, §5, p is P-regular.

For the rest, proceed as in Theorems 1 and 2 in Chapter 7, §7.

Lemma 8.8.7

If DP is σ-finite (Lemma 6), then CD is a Fubini map.

Proof

By Lemma 6,

(KP)p(KD)=0,DK.

Let Q=KD, so pQ=0, and CQ=CKCD; that is, CD=CKCQ and

X×YCQdp=pQ=0.

As KP,CK is a Fubini map. Thus by Lemma 2(ii), all reduces to proving the same for CQ.

Now, as pQ=0,Q is certainly σ-finite; so by Lemma 6,

(ZP)QZ,pZ=pQ=0.

Again CZ is a Fubini map; so

XYCZdndm=X×YCZdp=pZ=0.

As QZ, we have CQCZ, and so

XYCQdndm=X[YCQ(x,)dn]dmX[YCZ(x,)dn]dm=X×YCZdp=0.

Similarly,

YXCQdmdn=Y[XCQ(,y)dm]dn=0.

Thus setting

g(x)=YCQ(x,)dn and h(y)=XCQ(,y)dm,

we have

Xgdm=0=Yhdn.

Hence by Theorem 1(h) in §5, g=0 a.e. on X, and h=0 a.e. on Y. So g and h are "almost" measurable (Definition 2 of §3); i.e., CQ has Fubini property (a).

Similarly, one establishes (b), and (3) yields Fubini property (c), since

XYCQdndm=YXCQdmdn=X×YCQdp=0,

as required.

Theorem 8.8.3 (Fubini II)

Suppose f:X×YE is P-measurable on X×Y and satisfies condition (i) or (ii) of Theorem 2.

Then f is a Fubini map, provided f has σ-finite support, i.e., f vanishes outside some σ-finite set HX×Y.

Proof

First, let

f=i=1aiCDi(ai>0,DiP),

with f=0 on H (as above).

As f=ai0 on Ai, we must have DiH; so all Di are σ-finite. (Why?) Thus by Lemma 7, each CDi is a Fubini map, and so is f. (Why?)

If f is P-measurable and nonnegative, and f=0 on H, we can proceed as in Theorem 2, making all fk vanish on H. Then the fk and f are Fubini maps by what was shown above.

Finally, in case (ii), f=0 on H implies

f+=f=|f|=0 on H.

Thus f+,f, and f are Fubini maps by part (i) and the argument of Theorem 2.

Note 1. The σ-finite support is automatic if f is p-integrable (Corollary 1 in §5), or if p or both m and n are σ-finite (see Problem 3). The condition is also redundant if f is P-measurable (Theorem 2; see also Problem 4).

Note 2. By induction, our definitions and Theorems 2 and 3 extend to any finite number q of measure spaces

(Xi,Mi,mi),i=1,,q.

One writes

p=m1×m2

if q=2 and sets

m1×m2××mq+1=(m1××mq)×mq+1.

Theorems 2 and 3 with similar assumptions then state that the order of integrations is immaterial.

Note 3. Lebesgue measure in Eq can be treated as the product of q one- dimensional measures. Similarly for LS product measures (but this method is less general than that described in Problems 9 and 10 of Chapter 7, §9).

IV. Theorems 2(ii) and 3(ii) hold also for functions

f:X×YEn(Cn)

if Definitions 2 and 3 are modified as follows (so that they make sense for such maps): In Definition 2, set

g(x)=Yfxdn

if fx is n-integrable on Y, and g(x)=0 otherwise. Similarly for h(y). In Definition 3, replace "measurable" by "integrable."

For the proof of the theorems, apply Theorems 2(i) and 3(i) to |f|. This yields

YX|f|dmdn=XY|f|dndm=X×Y|f|dp.

Hence if one of these integrals is finite, f is p-integrable on X×Y, and so are its q components. The result then follows on noting that f is a Fubini map (in the modified sense) iff its components are. (Verify!) See also Problem 12 below.

V. In conclusion, note that integrals of the form

Dfdp(DP)

reduce to

X×YfCDdp.

Thus it suffices to consider integrals over X×Y.


8.8: Product Measures. Iterated Integrals is shared under a not declared license and was authored, remixed, and/or curated by LibreTexts.

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