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Mathematics LibreTexts

8.6.E: Problems on Integrability and Convergence Theorems

( \newcommand{\kernel}{\mathrm{null}\,}\)

Exercise 8.6.E.1

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

Exercise 8.6.E.2

(i) Show that if f:SE is bounded and m-measurable on A, with mA<, then f is m -integrable on A( Theorem 2) and
Af=cmA,
where inf f[A]csupf[A].
(ii) Prove that if f also has the Darboux property on A, then
(x0A)c=f(x0).
[Hint: Take g=1 in Theorem 3.]
(iii) What results if A=[a,b] and m= Lebesgue measure?

Exercise 8.6.E.3

Prove Theorem 4 assuming that the fn are measurable on A and that
(k)Afk>
instead of fn0.
[Hint: As {fn}, show that
(nk)Afn>.
If
(n)Afn=,
then
Af=limAfn=.
Otherwise,
(nk)|Afn|<;
so fn is integrable. (Why?) By Corollary 1 in §5, assume |fn|<. (Why?) Apply Theorem 4 to hn=fnfk(nk), considering two cases:
Ah< and Ah=.]

Exercise 8.6.E.4

Show that if fnf (pointwise) on AM, there are M-measurable maps Fnfn and Ff on A, with FnF (pointwise) on A, such that
AF=¯Af and AFn=¯Afn.
[Hint: By Lemma 2 of §5, fix measurable maps hf and hnfn with the same integrals. Let
Fn=infkn(hhk),n=1,2,,
and F=supnFnh. (Why?) Proceed.]

Exercise 8.6.E.5

For AM and any (even nonmeasurable) functions f,fn:SE, prove the following.
(i) If fnf( a.e. ) on A, then
¯Afn¯Af,
provided
(n)¯Afn>.
(ii) If fnf( a.e. ) on A, then
_Afn_Af,
provided
(n)_Afn<.
[Hint: Replace f,fn by F,Fn as in Problem 4. Then apply Problem 3 to Fn; thus obtain (i). For (ii), use (i) and Theorem 1(e) in §5. (All is orthodox; why?)]

Exercise 8.6.E.6

Show by examples that
(i) the conditions
¯Afn> and _Afn<
in Problem 5 are essential; and
(ii) Problem 5(i) fails for lower integrals. What about 5(ii)?
[Hints: (i) Let A=(0,1)E1,m= Lebesgue measure, fn= on (0,1n),fn=1 elsewhere.
(ii) Let M={E1,},mE1=1,m=0,fn=1 on (n,n),fn=0 elsewhere. If f=1 on A=E1, then fnf, but not
_Afn_Af.
Explain!]

Exercise 8.6.E.7

Given fn:SE and AM, let
gn=infknfk and hn=supknfk(n=1,2,).
Prove that
(i) ¯Alim_fnlim__Afn provided (n)¯Agn>; and
(ii) _A¯limfn¯lim_Afnprovided(n)_Ahn<.
[Hint: Apply Problem 5 to gn and hn.]
(iii) Give examples for which
¯Alim_fn¯limA¯Afn and _A¯limfnlim__Afn.
(See Note 2).

Exercise 8.6.E.8

Let fn0 on AM and fnf( a.e. ) on A. Let AX,XM.
Prove the following.
(i) If
¯Afn¯Af<,
then
¯Xfn¯Xf.
(ii) This fails for sign-changing fn.
[Hints: If (i) fails, then
lim_X¯Xfn<¯Xf or lim_X¯Xfn>¯Xf.
Find a subsequence of
{¯Xfn} or {¯AXfn}
contradicting Lemma 2.
(ii) Let m= Lebesgue measure; A=(0,1),X=(0,12),
fn={n on (0,12n],n on (112n,1[.

Exercise 8.6.E.9

9. (i) Show that if f and g are m-measurable and nonnegative on A, then
(a,b0)A(af+bg)=aAf+bAg.
(ii) If, in addition, Af< or Ag<, this formula holds for any a,bE1.
[Hint: Proceed as in Theorem 1.]

Exercise 8.6.E.10

10. If
f=n=1fn,
with all fn measurable and nonnegative on A, then
Af=n=1Afn.
[Hint: Apply Theorem 4 to the maps
gn=nk=1fkf.
Use Problem 9.]

Exercise 8.6.E.11

If
q=n=1A|fn|<
and the fn are m-measurable on A, then
n=1|fn|<(a.e.) on A
and f=n=1fn is m-integrable on A, with
Af=n=1Afn.
[Hint: Let g=n=1|fn|. By Problem 10,
Ag=n=1A|fn|=q<;
so g<(a.e.) on A. (Why?) Apply Theorem 5 and Note 1 to the maps
gn=nk=1fk;
note that |gn|g.]

Exercise 8.6.E.12

(Convergence in measure; see Problem 11(ii) of §3).
(i) Prove Riesz' theorem: If fnf in measure on AS, there is a subsequence {fnk} such that fnkf (almost uniformly), hence (a.e.), on A.
[Outline: Taking
σk=δk=2k,
pick, step by step, naturals
n1<n2<<nk<
and sets DkM such that (k)
mDk<2k
and
ρ(fnk,f)<2k
on ADk. (Explain!) Let
En=k=nDk,
mEn<21n.( Why?) Show that
(n)(k>n)ρ(fnk,f)<21n
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.]


8.6.E: Problems on Integrability and Convergence Theorems is shared under a CC BY 1.0 license and was authored, remixed, and/or curated by LibreTexts.

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