8.8.E: Problems on Product Measures and Fubini Theorems

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 $\left.A \subseteq \mathcal{P}^{*} . \text { Then let } f \text { be } p \text {-elementary, and so on. }\right]$