8.6.E: Problems on Integrability and Convergence Theorems
( \newcommand{\kernel}{\mathrm{null}\,}\)
Fill in the missing details in the proofs of this section.
(i) Show that if f:S→E∗ is bounded and m-measurable on A, with mA<∞, then f is m -integrable on A( Theorem 2) and
∫Af=c⋅mA,
where inf f[A]≤c≤supf[A].
(ii) Prove that if f also has the Darboux property on A, then
(∃x0∈A)c=f(x0).
[Hint: Take g=1 in Theorem 3.]
(iii) What results if A=[a,b] and m= Lebesgue measure?
Prove Theorem 4 assuming that the fn are measurable on A and that
(∃k)∫Afk>−∞
instead of fn≥0.
[Hint: As {fn}↑, show that
(∀n≥k)∫Afn>−∞.
If
(∃n)∫Afn=∞,
then
∫Af=lim∫Afn=∞.
Otherwise,
(∀n≥k)|∫Afn|<∞;
so fn is integrable. (Why?) By Corollary 1 in §5, assume |fn|<∞. (Why?) Apply Theorem 4 to hn=fn−fk(n≥k), considering two cases:
∫Ah<∞ and ∫Ah=∞.]
Show that if fn↗f (pointwise) on A∈M, there are M-measurable maps Fn≥fn and F≥f on A, with Fn↗F (pointwise) on A, such that
∫AF=¯∫Af and ∫AFn=¯∫Afn.
[Hint: By Lemma 2 of §5, fix measurable maps h≥f and hn≥fn with the same integrals. Let
Fn=infk≥n(h∧hk),n=1,2,…,
and F=supnFn≤h. (Why?) Proceed.]
For A∈M and any (even nonmeasurable) functions f,fn:S→E∗, prove the following.
(i) If fn↗f( a.e. ) on A, then
¯∫Afn↗¯∫Af,
provided
(∃n)¯∫Afn>−∞.
(ii) If fn↘f( 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?)]
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 fn→f, but not
∫_Afn→∫_Af.
Explain!]
Given fn:S→E∗ and A∈M, let
gn=infk≥nfk and hn=supk≥nfk(n=1,2,…).
Prove that
(i) ¯∫Alim_fn≤lim_∫_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¯limfn≠lim_∫_Afn.
(See Note 2).
Let fn≥0 on A∈M and fn→f( a.e. ) on A. Let A⊇X,X∈M.
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 {¯∫A−Xfn}
contradicting Lemma 2.
(ii) Let m= Lebesgue measure; A=(0,1),X=(0,12),
fn={n on (0,12n],−n on (1−12n,1[.
⇒9. (i) Show that if f and g are m-measurable and nonnegative on A, then
(∀a,b≥0)∫A(af+bg)=a∫Af+b∫Ag.
(ii) If, in addition, ∫Af<∞ or ∫Ag<∞, this formula holds for any a,b∈E1.
[Hint: Proceed as in Theorem 1.]
⇒10. If
f=∞∑n=1fn,
with all fn measurable and nonnegative on A, then
∫Af=∞∑n=1∫Afn.
[Hint: Apply Theorem 4 to the maps
gn=n∑k=1fk↗f.
Use Problem 9.]
If
q=∞∑n=1∫A|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=1∫Afn.
[Hint: Let g=∑∞n=1|fn|. By Problem 10,
∫Ag=∞∑n=1∫A|fn|=q<∞;
so g<∞(a.e.) on A. (Why?) Apply Theorem 5 and Note 1 to the maps
gn=n∑k=1fk;
note that |gn|≤g.]
(Convergence in measure; see Problem 11(ii) of §3).
(i) Prove Riesz' theorem: If fn→f in measure on A⊆S, there is a subsequence {fnk} such that fnk→f (almost uniformly), hence (a.e.), on A.
[Outline: Taking
σk=δk=2−k,
pick, step by step, naturals
n1<n2<⋯<nk<⋯
and sets Dk∈M such that (∀k)
mDk<2−k
and
ρ′(fnk,f)<2−k
on A−Dk. (Explain!) Let
En=∞⋃k=nDk,
mEn<21−n.( Why?) Show that
(∀n)(∀k>n)ρ′(fnk,f)<21−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]
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]
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.]