8.8: Product Measures. Iterated Integrals
- Page ID
- 32377
( \newcommand{\kernel}{\mathrm{null}\,}\)
Let (X,M,m) and (Y,N,n) be measure spaces, with X∈M and Y∈N. Let C be the family of all "rectangles," i.e., sets
A×B,
with A∈M,B∈N,mA<∞, and nB<∞.
Define a premeasure s:C→E1 by
s(A×B)=mA⋅nB,A×B∈C.
Let p∗ be the s-induced outer measure in X×Y and
p:P∗→E∗
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×Y→E∗ (extended-real).
I. We begin with some definitions.
Definitions
(1) Given a function f:X→Y→E∗ (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:X→E∗ by
g(x)=∫Yf(x,⋅)dn,
and set
∫X∫Yfdndm=∫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
∫Y∫Xfdmdn=∫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 x∈X−Q, mQ=0);fy is m-measurable on X for y∈Y−Q′,nQ′=0; and
(c) the iterated integrals above satisfy
∫X∫Yfdndm=∫Y∫Xfdmdn=∫X×Yfdp
(the main point).
For monotone sequences
fk:X×Y→E∗(k=1,2,…),
we now obtain the following lemma.
Lemma 8.8.1
If 0≤fk↗f (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,
0≤fk(x,⋅)↗f(x,⋅)
pointwise on Y. Thus by Theorem 4 in §6,
∫Yfk(x,⋅)↗∫Yf(x,⋅)dn,
i.e., gk↗g (pointwise) on X, with g as in Definition 2.
Again, by Theorem 4 of §6,
∫Xgkdm↗∫Xgdm;
or by Definition 2,
∫X∫Yfdndm=limk→∞∫X∫Yfkdndm.
Similarly for
∫Y∫Xfdmdn
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 x∈X−Qk (mQk=0). Let
Q=∞⋃k=1Qk;
so mQ=0, and all fk(x,⋅) are n-measurable on Y, for x∈X−Q. Hence so is
f(x,⋅)=limk→∞fk(x,⋅).
Similarly for f(⋅,y). Thus f satisfies (b).
Property (a) follows from gk→g and hk→h.◻
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,b≥0.
(ii) If, in addition,
∫X×Yf1dp<∞ or ∫X×Yf2dp<∞,
then f1−f2 is a Fubini map, too
Lemma 8.8.3
Let f=∑∞i=1fi (pointwise), with fi≥0 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
∫X∫Yfdndm=∞∑i=1∫X∫Yfidndm
and
∫Y∫Xfdmdn=∞∑i=1∫Y∫Xfidmdn.
II. By Theorem 4 of Chapter 7, §3, the family C (see above) is a semiring, being the product of two rings,
{A∈M|mA<∞} and {B∈N|nB<∞}.
(Verify!) Thus using Theorem 2 in Chapter 7, §6, we now show that p is an extension of s:C→E1.
Theorem 8.8.1
The product premeasure s is σ-additive on the semiring C. Hence
(i) C⊆P∗ and p=s<∞ on C, and
(ii) the characteristic function CD of any set D∈C is a Fubini map.
- Proof
-
Let D=A×B∈C; 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=CA⋅nB is M-simple on X, with
∫X∫YCDdndm=∫Xgdm=nB∫XCAdm=nB⋅mA=sD.
Similarly for CD(⋅,y), and
h(y)=∫XCD(⋅,y)dm.
Thus CD has Fubini properties (a) and (b), and for every D∈C
∫X∫YCDdndm=∫Y∫XCDdmdn=sD.
To prove σ-additivity, let
D=∞⋃i=1Di (disjoint), Di∈C;
so
CD=∞∑i=1CDi.
(Why?) As shown above, each CDi has Fubini properties (a) and (b); so by (1) and Lemma 3,
sD=∫X∫YCDdndm=∞∑i=1∫X∫YCDidndm=∞∑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 C⊆P⊆P∗).
Lemma 8.8.4
P is the least set family R such that
(i) R⊇C;
(ii) R is closed under countable disjoint unions; and
(iii) H−D∈R if D∈R and D⊆H,H∈C.
This is simply Theorem 3 in Chapter 7, §3, with changed notation.
Lemma 8.8.5
If D∈P (σ-generated by C), then CD is a Fubini map.
- Proof
-
Let R be the family of all D∈P such that CD is a Fubini map. We shall show that R satisfies (i)-(iii) of Lemma 4, and so P⊆R.
(ii) Let
D=∞⋃i=1Di (disjoint),Di∈R.
Then
CD=∞∑i=1CDi,
and each CDi is a Fubini map. Hence so is CD by Lemma 3. Thus D∈R, proving (ii).
(iii) We must show that CH−D is a Fubini map if CD is and if D⊆H,H∈C. Now, D⊆H implies
CH−D=CH−CD.
(Why?) Also, by Theorem 1, H∈C implies
∫X×YCHdp=pH=sH<∞,
and CH is a Fubini map. So is CD by assumption. So also is
CH−D=CH−CD
by Lemma 2(ii). Thus H−D∈R, proving (iii).
By Lemma 4, then, P⊆R. Hence (∀D∈P)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×Y→E∗ is P-measurable on X×Y (P as above) rom. Then f is a Fubini map if either
(i) f≥0 on X×Y, or
(ii) one of
∫X×Y|f|dp,∫X∫Y|f|dndm,or∫Y∫X|f|dmdn
is finite.
In both cases,
∫X∫Yfdndm=∫Y∫Xfdmdn=∫X×Yfdp.
- Proof
-
First, let
f=∞∑i=1aiCDi(ai≥0,Di∈P),
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 f≥0. By Lemma 2 in §2,
f=limk→∞fk on X×Y
for some sequence {fk}↑ of P-elementary maps, fk≥0. 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 D∈P∗ be σ-finite, i.e.,
D=∞⋃i=1Di (disjoint)
for some Di∈P∗, with pDi<∞ (i=1,2,…).
Then there is aK∈P such that p(K−D)=0 and D⊆K.
- 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 D∈P∗ is σ-finite (Lemma 6), then CD is a Fubini map.
- Proof
-
By Lemma 6,
(∃K∈P)p(K−D)=0,D⊆K.
Let Q=K−D, so pQ=0, and CQ=CK−CD; that is, CD=CK−CQ and
∫X×YCQdp=pQ=0.
As K∈P,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,
(∃Z∈P)Q⊆Z,pZ=pQ=0.
Again CZ is a Fubini map; so
∫X∫YCZdndm=∫X×YCZdp=pZ=0.
As Q⊆Z, we have CQ≤CZ, and so
∫X∫YCQdndm=∫X[∫YCQ(x,⋅)dn]dm≤∫X[∫YCZ(x,⋅)dn]dm=∫X×YCZdp=0.
Similarly,
∫Y∫XCQdmdn=∫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
∫X∫YCQdndm=∫Y∫XCQdmdn=∫X×YCQdp=0,
as required.◻
Theorem 8.8.3 (Fubini II)
Suppose f:X×Y→E∗ 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 H⊆X×Y.
- Proof
-
First, let
f=∞∑i=1aiCDi(ai>0,Di∈P∗),
with f=0 on −H (as above).
As f=ai≠0 on Ai, we must have Di⊆H; 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×Y→En(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
∫Y∫X|f|dmdn=∫X∫Y|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(D∈P∗)
reduce to
∫X×Yf⋅CDdp.
Thus it suffices to consider integrals over X×Y.