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# 8.8.E: Problems on Product Measures and Fubini Theorems

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## Exercise $$\PageIndex{1}$$

Prove Lemmas 2 and 3.

## Exercise $$\PageIndex{1'}$$

Show that $$\{A \in \mathcal{M} | m A<\infty\}$$ is a set ring.

## Exercise $$\PageIndex{2}$$

Fill in all proof details in Theorems 1 to 3.

## Exercise $$\PageIndex{2'}$$

Do the same for Lemmas 5 to 7.

## Exercise $$\PageIndex{3}$$

Prove that if $$m$$ and $$n$$ are $$\sigma$$-finite, so is $$p=m \times n .$$ Disprove the converse by an example.
[Hint: $$\left(\cup_{i} A_{i}\right) \times\left(U_{j} B_{j}\right)=U_{i, j}\left(A_{i} \times B_{j}\right)$$. Verify!]

## Exercise $$\PageIndex{4}$$

Prove the following.
(i) Each $$D \in \mathcal{P}$$ (as in the text) is (p) $$\sigma$$-finite.
(ii) All $$\mathcal{P}$$-measurable maps $$f: X \times Y \rightarrow E^{*}$$ have $$\sigma$$-finite support.
[Hints: (i) Use Problem $$14(\mathrm{b})$$ from Chapter 7, §3. (ii) Use (i) for $$\mathcal{P}$$-elementary and nonnegative maps first. $$]$$

## Exercise $$\PageIndex{5}$$

(i) Find $$D \in \mathcal{P}^{*}$$ and $$x \in X$$ such that $$C_{D}(x, \cdot)$$ is not $$n$$-measurable on $$Y .$$ Does this contradict Lemma $$7 ?$$
[Hint: Let $$m=n=$$ Lebesgue measure in $$E^{1} ; D=\{x\} \times Q,$$ with $$Q$$ non-measurable. $$]$$
(ii) Which $$\mathcal{C}$$-sets have nonzero measure if $$X=Y=E^{1}, m^{*}$$ is as in Problem $$2(b)$$ of Chapter $$7, §5(\text { with } S=X),$$ and $$n$$ is Lebesgue measure?

## Exercise $$\PageIndex{5'}$$

Let $$m=n=$$ Lebesgue measure in $$[0,1]=X=Y .$$ Let
$f_{k}=\left\{\begin{array}{ll}{k(k+1)} & {\text { on }\left(\frac{1}{k+1}, \frac{1}{k}\right] \text { and }} \\ {0} & {\text { elsewhere. }}\end{array}\right.$
Let
$f(x, y)=\sum_{k=1}^{\infty}\left[f_{k}(x)-f_{k+1}(x)\right] f_{k}(y) ;$
the series converges. (Why?) Show that
(i) $$(\forall k) \int_{X} f_{k}=1$$;
(ii) $$\int_{X} \int_{Y} f d n d m=1 \neq 0=\int_{Y} \int_{X} f d m d n$$.
What is wrong? Is $$f$$ $$\mathcal{P}$$-measurable?
[Hint: Explore
$\left.\int_{X} \int_{Y}|f| d n d m .\right]$

## Exercise $$\PageIndex{6}$$

Let $$X=Y=[0,1], m$$ as in Example $$(\mathrm{c})$$ of Chapter $$7, §6,(S=X)$$ and $$n=$$ Lebesgue measure in $$Y .$$
(i) Show that $$p=m \times n$$ is a topological measure under the standard metric in $$E^{2} .$$
(ii) Prove that $$D=\{(x, y) \in X \times Y | x=y\} \in \mathcal{P}^{*}$$.
(iii) Describe $$\mathcal{C}$$.
[Hints: (i) Any subinterval of $$X \times Y$$ is in $$\mathcal{P}^{*} ;$$ (ii) $$D$$ is closed. Verify!]

## Exercise $$\PageIndex{7}$$

Continuing Problem $$6,$$ let $$f=C_{D}$$.
(i) Show that
$\int_{Y} \int_{X} f d n d m=0 \neq 1=\int_{Y} \int_{X} f d m d n .$
What is wrong?
[Hint: $$D$$ is not $$\sigma$$-finite; for if
$D=\bigcup_{i=1}^{\infty} D_{i} ,$
at least one $$D_{\mathrm{i}}$$ is uncountable and has no finite basic covering values (why?), so $$p^{*} D_{\mathrm{i}}=\infty .$$]
(ii) Compute $$p^{*}\{(x, 0) | x \in X\}$$ and $$p^{*}\{(0, y) | y \in Y\}$$.

## Exercise $$\PageIndex{8}$$

Show that $$D \in \mathcal{P}^{*}$$ is $$\sigma$$-finite iff
$D \subseteq \bigcup_{i=1}^{\infty} D_{i}(\text {disjoint})$
for some sets $$D_{i} \in \mathcal{C}$$.
[Hint: First let $$\left.p^{*} D<\infty . \text { Use Corollary } 1 \text { from Chapter } 7, §1 .\right]$$

## Exercise $$\PageIndex{9}$$

Given $$D \in \mathcal{P}, a \in X,$$ and $$b \in Y,$$ let
$D_{a}=\{y \in Y |(a, y) \in D\}$
and
$D^{b}=\{x \in X |(x, b) \in D\} .$
(See Figure $$\left.34 \text { for } X=Y=E^{1} .\right)$$
Prove that
(i) $$D_{a} \in \mathcal{N}, D^{b} \in \mathcal{M}$$;
(ii) $$C_{D}(a, \cdot)=C_{D_{a}}, n D_{a}=\int_{Y} C_{D}(a, \cdot) d n, m D^{b}=\int_{X} C_{D}(\cdot, b) d m$$.
[Hint: Let
$H=\left\{(x, y) \in E^{2} | 0 \leq y<f(x)\right\}$
Show that $$\mathcal{R}$$ is a $$\sigma$$-ring $$\supseteq C .$$ Hence $$\mathcal{R} \supseteq \mathcal{P} ; D \in \mathcal{R} ; D_{a} \in \mathcal{N} .$$ Similariy for $$D^{b} .$$]

## Exercise $$\PageIndex{10}$$

$$\Rightarrow 10$$. Let $$m=n=$$ Lebesgue measure in $$E^{1}=X=Y .$$ Let $$f: E^{1} \rightarrow[0, \infty)$$ be $$m$$-mensurable on $$X .$$ Let
$H=\left\{(x, y) \in E^{2} | 0 \leq y<f(x)\right\}$
and
$G=\left\{(x, y) \in E^{2} | y=f(x, y)\right\}$
(the "graph" of $$f$$ ). Prove that
(i) $$H \in \mathcal{P}^{*}$$ and
$p H=\int_{X} f d m$
(="the area under f")
(ii) $$G \in \mathcal{P}^{*}$$ and $$p G=0$$.
[Hints: (i) First take $$f=C_{D},$$ and elementary and nonnegative maps. Then use Lemma 2 in §2 (last clause). Fix elementary and nonnegative maps $$f_{k} \nearrow f,$$ assuming $$\left.f_{k}<f \text { (if not, replace } f_{k} \text { by }\left(1-\frac{1}{k}\right) f_{k}\right) .$$ Let
$H_{k}=\left\{(x, y) | 0 \leq y<f_{k}(x)\right\} .$
Show that $$H_{k} \nearrow H \in \mathcal{P}^{*}$$.
(ii) Set
$\phi(x, y)=y-f(x) .$
Using Corollary 4 of §1, show that $$\phi$$ is $$p$$-measurable on $$E^{2} ;$$ so $$G=E^{2}(\phi=0) \in \mathcal{P}^{*}$$. Dropping a null set (Lemma $$6),$$ assume $$G \in \mathcal{P} .$$ By Problem 9 (ii),
$\left(\forall x \in E^{1}\right) \quad \int_{Y} C_{G}(x, \cdot) d n=n G_{x}=0 ,$
as $$\left.G_{x}=\{f(x)\}, \text { a singleton. }\right]$$

## Exercise $$\PageIndex{11}$$

Let
$f(x, y)=\phi_{1}(x) \phi_{2}(y) .$
Prove that if $$\phi_{1}$$ is $$m$$-integrable on $$X$$ and $$\phi_{2}$$ is $$n$$-integrable on $$Y,$$ then $$f$$ is $$p$$-integrable on $$X \times Y$$ and
$\int_{X \times Y} f d p=\int_{X} \phi_{1} \cdot \int_{Y} \phi_{2} .$

## Exercise $$\PageIndex{*12}$$

Prove Theorem $$3(\text {ii) for } f: X \times Y \rightarrow E(E\text { complete) }$$.
[Outline: If $$f$$ is $$\mathcal{P}^{*}$$-simple, use Lemma 7 above and Theorem 2 in §7.
If
$f=\sum_{k=1}^{\infty} a_{k} C_{D_{k}}, \quad D_{k} \in \mathcal{P}^{*} ,$
let
$H_{k}=\bigcup_{i=1}^{k} D_{i}$
and $$f_{k}=f C_{H_{k}},$$ so the $$f_{k}$$ are $$\mathcal{P}^{*}$$-simple (hence Fubini maps), and $$f_{k} \rightarrow f$$ (point-wise) on $$X \times Y,$$ with $$\left|f_{k}\right| \leq|f|$$ and
$\int_{X \times Y}|f| d p<\infty$
(by assumption). Now use Theorem 5 from §6.
Let now $$f$$ be $$\mathcal{P}^{*}$$-measurable; so
$f=\lim _{k \rightarrow \infty} f_{k} \text { (uniformly) }$
for some $$\left.\mathcal{P}^{*} \text {-elementary maps } g_{k} \text { (Theorem } 3 \text { in } §1\right) .$$ By assumption, $$f=f C_{H}(H$$ $$\sigma$$-finite); so we may assume $$g_{k}=g_{k} C_{H} .$$ Then as shown above, all $$g_{k}$$ are Fubini maps. So is $$f$$ by Lemma 1 in §7 (verify!), provided $$H \subseteq D$$ for some $$D \in \mathcal{C} .$$
In the general case, by Problem 8 ,
$H \subseteq \bigcup_{i} D_{i}(\text {disjoint}), D_{i} \in \mathcal{C} .$
Let $$H_{i}=H \cap D_{i} .$$ By the previous step, each $$f C_{H_{i}}$$ is a Fubini map; so is
$f_{k}=\sum_{i=1}^{k} f C_{H_{i}}$
(why?), hence so is $$f=\lim _{k \rightarrow \infty} f_{k},$$ by Theorem 5 of §6. (Verify!)]

## Exercise $$\PageIndex{13}$$

Let $$m=$$ Lebesgue measure in $$E^{1}, p=$$ Lebesgue measure in $$E^{s}, X= (0, \infty),$$ and
$Y=\left\{\bar{y} \in E^{s}|| \bar{y} |=1\right\} .$
Given $$\bar{x} \in E^{s}-\{\overline{0}\},$$ let
$r=|\bar{x}| \text { and } \bar{u}=\frac{\bar{x}}{r} \in Y .$
Call $$r$$ and $$\bar{u}$$ the polar coordinates of $$\bar{x} \neq \overline{0}$$.
If $$D \subseteq Y,$$ set
$n^{*} D=s \cdot p^{*}\{r \bar{u} | \bar{u} \in D, 0<r \leq 1\} .$
Show that $$n^{*}$$ is an outer measure in $$Y ;$$ so it induces a measure $$n$$ in $$Y .$$
Then prove that
$\int_{E^{s}} f d p=\int_{X} r^{s-1} d m(r) \int_{Y} f(r \bar{u}) d n(\bar{u})$
if $$f$$ is $$p$$-measurable and nonnegative on $$E^{s} .$$
[Hint: Start with $$f=C_{A},$$
$A=\{r \bar{u} | \bar{u} \in H, a<r<b\} ,$
for some open set $$\left.H \subseteq Y \text { (subspace of } E^{s}\right) .$$ Next, let $$A \in \mathcal{B}(\text { Borel set in } Y) ;$$ then $$\left.A \subseteq \mathcal{P}^{*} . \text { Then let } f \text { be } p \text {-elementary, and so on. }\right]$$