8.6.E: Problems on Integrability and Convergence Theorems

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
$$\left\{\overline{\int}_{X} f_{n}\right\} \text { or }\left\{\overline{\int}_{A-X} f_{n}\right\}$$
contradicting Lemma 2.
(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.]