8.8.E: Problems on Product Measures and Fubini Theorems
Prove Lemmas 2 and 3.
Show that \(\{A \in \mathcal{M} | m A<\infty\}\) is a set ring.
Fill in all proof details in Theorems 1 to 3.
Do the same for Lemmas 5 to 7.
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!]
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. \(]\)
(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?
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
\[
\left.\int_{X} \int_{Y}|f| d n d m .\right]
\]
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!]
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 \(p^{*}\{(0, y) | y \in Y\}\).
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 \(\left.p^{*} D<\infty . \text { Use Corollary } 1 \text { from Chapter } 7, §1 .\right]\)
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 \(D^{b} .\)]
\(\Rightarrow 10\). 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 \(\left.G_{x}=\{f(x)\}, \text { a singleton. }\right]\)
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
\[
\int_{X \times Y} f d p=\int_{X} \phi_{1} \cdot \int_{Y} \phi_{2} .
\]
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!)]
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 \(\left.A \subseteq \mathcal{P}^{*} . \text { Then let } f \text { be } p \text {-elementary, and so on. }\right]\)