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# 8.6.E: Problems on Integrability and Convergence Theorems

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

Fill in the missing details in the proofs of this section.

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

(i) Show that if $$f: S \rightarrow E^{*}$$ is bounded and $$m$$-measurable on $$A,$$ with $$m A<\infty,$$ then $$f$$ is $$m$$ -integrable on $$A(\text { Theorem } 2)$$ and
$\int_{A} f=c \cdot m A ,$
where inf $$f[A] \leq c \leq \sup f[A]$$.
(ii) Prove that if $$f$$ also has the Darboux property on $$A,$$ then
$\left(\exists x_{0} \in A\right) \quad c=f\left(x_{0}\right) .$
[Hint: Take $$g=1 \text { in Theorem } 3 .]$$
(iii) What results if $$A=[a, b]$$ and $$m=$$ Lebesgue measure?

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

Prove Theorem 4 assuming that the $$f_{n}$$ are measurable on $$A$$ and that
$(\exists k) \quad \int_{A} f_{k}>-\infty$
instead of $$f_{n} \geq 0$$.
[Hint: As $$\left\{f_{n}\right\} \uparrow$$, show that
$(\forall n \geq k) \quad \int_{A} f_{n}>-\infty .$
If
$(\exists n) \quad \int_{A} f_{n}=\infty ,$
then
$\int_{A} f=\lim \int_{A} f_{n}=\infty .$
Otherwise,
$(\forall n \geq k) \quad\left|\int_{A} f_{n}\right|<\infty ;$
so $$f_{n}$$ is integrable. (Why?) By Corollary 1 in §5, assume $$\left|f_{n}\right|<\infty .$$ (Why?) Apply Theorem 4 to $$h_{n}=f_{n}-f_{k}(n \geq k),$$ considering two cases:
$\left.\int_{A} h<\infty \text { and } \int_{A} h=\infty .\right]$

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

Show that if $$f_{n} \nearrow f$$ (pointwise) on $$A \in \mathcal{M},$$ there are $$\mathcal{M}$$-measurable maps $$F_{n} \geq f_{n}$$ and $$F \geq f$$ on $$A,$$ with $$F_{n} \nearrow F$$ (pointwise) on $$A,$$ such that
$\int_{A} F=\overline{\int}_{A} f \text { and } \int_{A} F_{n}=\overline{\int}_{A} f_{n} .$
[Hint: By Lemma 2 of §5, fix measurable maps $$h \geq f$$ and $$h_{n} \geq f_{n}$$ with the same integrals. Let
$F_{n}=\inf _{k \geq n}\left(h \wedge h_{k}\right), \quad n=1,2, \ldots ,$
and $$F=\sup _{n} F_{n} \leq h .$$ (Why?) Proceed.]

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

For $$A \in \mathcal{M}$$ and any (even nonmeasurable) functions $$f, f_{n}: S \rightarrow E^{*},$$ prove the following.
(i) If $$f_{n} \nearrow f(\text { a.e. })$$ on $$A,$$ then
$\overline{\int}_{A} f_{n} \nearrow \overline{\int}_{A} f ,$
provided
$(\exists n) \quad \overline{\int}_{A} f_{n}>-\infty .$
(ii) If $$f_{n} \searrow f(\text { a.e. })$$ on $$A,$$ then
$\underline{\int}_{A} f_{n} \searrow \underline{\int}_{A} f ,$
provided
$(\exists n) \quad \underline{\int}_{A} f_{n}<\infty .$
[Hint: Replace $$f, f_{n}$$ by $$F, F_{n}$$ as in Problem $$4 .$$ Then apply Problem 3 to $$F_{n} ;$$ thus obtain (i). For (ii), use (i) and Theorem $$1\left(\mathrm{e}^{\prime}\right)$$ in §5. (All is orthodox; why?)]

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

Show by examples that
(i) the conditions
$\overline{\int}_{A} f_{n}>-\infty \text { and } \underline{\int}_{A} f_{n}<\infty$
in Problem 5 are essential; and
(ii) Problem $$5(\mathrm{i})$$ fails for lower integrals. What about $$5(\mathrm{ii}) ?$$
[Hints: (i) Let $$A=(0,1) \subset E^{1}, m=$$ Lebesgue measure, $$f_{n}=-\infty$$ on $$\left(0, \frac{1}{n}\right), f_{n}=1$$ elsewhere.
(ii) Let $$\mathcal{M}=\left\{E^{1}, \emptyset\right\}, m E^{1}=1, m \emptyset=0, f_{n}=1$$ on $$(-n, n), f_{n}=0$$ elsewhere. If $$f=1$$ on $$A=E^{1},$$ then $$f_{n} \rightarrow f,$$ but not
$\underline{\int}_{A} f_{n} \rightarrow \underline{\int}_{A} f .$
Explain!]

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

Given $$f_{n}: S \rightarrow E^{*}$$ and $$A \in \mathcal{M},$$ let
$g_{n}=\inf _{k \geq n} f_{k} \text { and } h_{n}=\sup _{k \geq n} f_{k} \quad(n=1,2, \ldots) .$
Prove that
(i) $$\overline{\int}_{A} \underline{\lim} f_{n} \leq \underline{\lim} \underline{\int}_{A} f_{n}$$ provided $$(\exists n) \overline{\int}_{A} g_{n}>-\infty ;$$ and
(ii) $$\underline{\int}_{A} \overline{\lim } f_{n} \leq \overline{\lim } \underline{\int}_{A} f_{n} \text{provided}(\exists n) \underline{\int}_{A} h_{n}<\infty$$.
[Hint: Apply Problem 5 to $$\left.g_{n} \text { and } h_{n} .\right]$$
(iii) Give examples for which
$\overline{\int}_{A} \underline{\lim} f_{n} \neq \overline{\lim}_{A} \overline{\int}_{A} f_{n} \text { and } \underline{\int}_{A} \overline{\lim } f_{n} \neq \underline{\lim } \underline{\int}_{A} f_{n} .$
(See Note 2).

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

Let $$f_{n} \geq 0$$ on $$A \in \mathcal{M}$$ and $$f_{n} \rightarrow f(\text { a.e. })$$ on $$A .$$ Let $$A \supseteq X, X \in \mathcal{M} .$$
Prove the following.
(i) If
$\overline{\int}_{A} f_{n} \rightarrow \overline{\int}_{A} f<\infty ,$
then
$\overline{\int}_{X} f_{n} \rightarrow \overline{\int}_{X} f .$
(ii) This fails for sign-changing $$f_{n}$$.
[Hints: If (i) fails, then
$\underline{\lim} _{X} \overline{\int}_{X} f_{n}<\overline{\int}_{X} f \text { or } \underline{\lim}_{X} \overline{\int}_{X} f_{n}>\overline{\int}_{X} f .$
Find a subsequence of
$\left\{\overline{\int}_{X} f_{n}\right\} \text { or }\left\{\overline{\int}_{A-X} f_{n}\right\}$
(ii) Let $$m=$$ Lebesgue measure; $$A=(0,1), X=\left(0, \frac{1}{2}\right)$$,
$f_{n}=\left\{\begin{array}{ll}{n} & {\text { on }\left(0, \frac{1}{2 n}\right],} \\ {-n} & {\text { on }\left(1-\frac{1}{2 n}, 1\right[ .}\end{array}\right.$

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

$$\Rightarrow 9$$. (i) Show that if $$f$$ and $$g$$ are $$m$$-measurable and nonnegative on $$A,$$ then
$(\forall a, b \geq 0) \quad \int_{A}(a f+b g)=a \int_{A} f+b \int_{A} g .$
(ii) If, in addition, $$\int_{A} f<\infty$$ or $$\int_{A} g<\infty,$$ this formula holds for any $$a, b \in E^{1} .$$
[Hint: Proceed as in Theorem 1.]

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

$$\Rightarrow 10$$. If
$f=\sum_{n=1}^{\infty} f_{n} ,$
with all $$f_{n}$$ measurable and nonnegative on $$A,$$ then
$\int_{A} f=\sum_{n=1}^{\infty} \int_{A} f_{n} .$
[Hint: Apply Theorem 4 to the maps
$g_{n}=\sum_{k=1}^{n} f_{k} \nearrow f .$
Use Problem 9.]

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

If
$q=\sum_{n=1}^{\infty} \int_{A}\left|f_{n}\right|<\infty$
and the $$f_{n}$$ are $$m$$-measurable on $$A,$$ then
$\sum_{n=1}^{\infty}\left|f_{n}\right|<\infty(a . e .) \text { on } A$
and $$f=\sum_{n=1}^{\infty} f_{n}$$ is $$m$$-integrable on $$A,$$ with
$\int_{A} f=\sum_{n=1}^{\infty} \int_{A} f_{n} .$
[Hint: Let $$g=\sum_{n=1}^{\infty}\left|f_{n}\right| .$$ By Problem 10,
$\int_{A} g=\sum_{n=1}^{\infty} \int_{A}\left|f_{n}\right|=q<\infty ;$
so $$g<\infty(a . e .)$$ on $$A .$$ (Why?) Apply Theorem 5 and Note 1 to the maps
$g_{n}=\sum_{k=1}^{n} f_{k} ;$
note that $$\left.\left|g_{n}\right| \leq g .\right]$$

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

(Convergence in measure; see Problem 11(ii) of §3).
(i) Prove Riesz' theorem: If $$f_{n} \rightarrow f$$ in measure on $$A \subseteq S$$, there is a subsequence $$\left\{f_{n_{k}}\right\}$$ such that $$f_{n_{k}} \rightarrow f$$ (almost uniformly), hence (a.e.), on $$A$$.
[Outline: Taking
$\sigma_{k}=\delta_{k}=2^{-k} ,$
pick, step by step, naturals
$n_{1}<n_{2}<\cdots<n_{k}<\cdots$
and sets $$D_{k} \in \mathcal{M}$$ such that $$(\forall k)$$
$m D_{k}<2^{-k}$
and
$\rho^{\prime}\left(f_{n_{k}}, f\right)<2^{-k}$
on $$A-D_{k} .$$ (Explain!) Let
$E_{n}=\bigcup_{k=n}^{\infty} D_{k} ,$
$$m E_{n}<2^{1-n} .($$ Why?) Show that
$(\forall n)(\forall k>n) \quad \rho^{\prime}\left(f_{n_{k}}, f\right)<2^{1-n}$
on $$\left.A-E_{n} . \text { Use Problem } 11 \text { in } §3 .\right]$$
(ii) For maps $$f_{n}: S \rightarrow E$$ and $$g: S \rightarrow E^{1}$$ deduce that if
$f_{n} \rightarrow f$
in measure on $$A$$ and
$(\forall n) \quad\left|f_{n}\right| \leq g(\text { a.e. }) \text { on } A ,$
then
$|f| \leq g(\text { a.e. }) \text { on } A .$
$$\left[\text { Hint: } f_{n_{k}} \rightarrow f(a . e .) \text { on } A .\right]$$

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

Continuing Problem $$12(\text { ii }),$$ let
$f_{n} \rightarrow f$
in measure on $$A \in \mathcal{M}\left(f_{n}: S \rightarrow E\right)$$ and
$(\forall n) \quad\left|f_{n}\right| \leq g(\mathrm{a.e.}) \text { on } A ,$
with
$\overline{\int_{A}} g<\infty .$
Prove that
$\lim _{n \rightarrow \infty} \overline{\int}_{A}\left|f_{n}-f\right|=0 .$
Does
$\overline{\int}_{A} f_{n} \rightarrow \overline{\int}_{A} f ?$
[Outline: From Corollary 1 of §5, infer that $$g=0$$ on $$A-C,$$ where
$C=\bigcup_{k=1}^{\infty} C_{k}(\text {disjoint}) ,$
$$m C_{k}<\infty .$$ (We may assume $$g \mathcal{M}$$-measurable on $$A .$$ Why?) Also,
$\infty>\int_{A} g=\int_{A-C} g+\int_{C} g=0+\sum_{k=1}^{\infty} \int_{C_{k}} g ;$
so the series converges. Hence
$(\forall \varepsilon>0)(\exists p) \quad \int_{A} g-\varepsilon<\sum_{k=1}^{p} \int_{C_{k}} g=\int_{H} g ,$
where
$H=\bigcup_{k=1}^{p} C_{k} \in \mathcal{M}$
and $$m H<\infty .$$ As $$\left|f_{n}-f\right| \leq 2 g(\text { a.e. }),$$ we get
$\text { (1) } \underline{\int}_{A}\left|f_{n}-f\right| \leq \overline{\int}_{A}\left|f_{n}-f\right| \leq \overline{\int}_{H}\left|f_{n}-f\right|+\int_{A-H} 2 g<\overline{\int_{H}}\left|f_{n}-f\right|+2 \varepsilon .$
(Explain!)
As $$m H<\infty,$$ we can fix $$\sigma>0$$ with
$\sigma \cdot m H<\varepsilon .$
Also, by Theorem $$6,$$ fix $$\delta$$ such that
$2 \int_{X} g<\varepsilon$
whenever $$A \supseteq X, X \in \mathcal{M}$$ and $$m X<\delta$$.
As $$f_{n} \rightarrow f$$ in measure on $$H,$$ we find $$\mathcal{M}$$-sets $$D_{n} \subseteq H$$ such that
$\left(\forall n>n_{0}\right) \quad m D_{n}<\delta$
and
$\left|f_{n}-f\right|<\sigma \text { on } A_{n}=H-D_{n} .$
(We may use the standard metric, as $$|f|$$ and $$\left|f_{n}\right|<\infty$$ a.e. Why?) Thus from $$(1),$$ we get
\begin{aligned} \overline{\int}_{A}\left|f_{n}-f\right| & \leq \overline{\int_{H}}\left|f_{n}-f\right|+2 \varepsilon \\ &=\overline{\int}_{A_{n}}\left|f_{n}-f\right|+\overline{\int}_{D_{n}}\left|f_{n}-f\right|+2 \varepsilon \\ &<\overline{\int}_{A_{n}}\left|f_{n}-f\right|+3 \varepsilon \\ & \leq \sigma \cdot m H+3 \varepsilon<4 \varepsilon \end{aligned}
for $$n>n_{0} .$$ (Explain!) Hence
$\lim \overline{\int_{A}}\left|f_{n}-f\right|=0 .$
See also Problem 7 in §5 and Note 1 of §6 (for measurable functions) as regards
$\left.\lim \overline{\int_{A}} f_{n} \cdot\right]$

## Exercise $$\PageIndex{14}$$

Do Problem 12 in §3 (Lebesgue-Egorov theorems) for $$T=E,$$ assuming
$(\forall n) \quad\left|f_{n}\right| \leq g(a . e .) \text { on } A ,$
with
$\int_{A} g<\infty$
(instead of $$m A<\infty)$$.
[Hint: With $$H_{i}(k)$$ as before, it suffices that
$\lim _{i \rightarrow \infty} m\left(A-H_{i}(k)\right)=0 .$
(Why?) Verify that
$(\forall n) \quad \rho^{\prime}\left(f_{n}, f\right)=\left|f_{n}-f\right| \leq 2 g(a . e .) \text { on } A ,$
and
$(\forall i, k) \quad A-H_{i}(k) \subseteq A\left(2 g \geq \frac{1}{k}\right) \cup Q(m Q=0) .$
Infer that
$(\forall i, k) \quad m\left(A-H_{i}(k)\right)<\infty.$
Now, as $$(\forall k) H_{i}(k) \searrow \emptyset$$ (why?), right continuity applies.]