Integration: Exercises
Arun Ram
Department of Mathematics and Statistics
University of Melbourne
Parkville, VIC 3010 Australia
aram@unimelb.edu.au
Last updates: 5 March 2011
Integration: Exercises

Does there exist an infinite $\sigma $algebra which has only countably many members?
 Prove an analogue of [Ru, Theorem 1.8] for $n$ functions.
 Prove that if $f$ is a real function on a measurable space $X$
such that $\left\{x\phantom{\rule{0.2em}{0ex}}\right\phantom{\rule{0.2em}{0ex}}f\left(x\right)\ge r\}$
is measurable for every rational $r$, then $f$ is
measurable.
 Let $\left\{{a}_{n}\right\}$ and
$\left\{{b}_{n}\right\}$ be sequences in
$[\infty ,\infty ]$
and prove that
 (a) $\underset{n\to \infty}{\mathrm{limsup}}}\phantom{\rule{0.2em}{0ex}}({a}_{n})={\displaystyle \underset{n\to \infty}{\mathrm{liminf}}}\phantom{\rule{0.2em}{0ex}}{a}_{n$.
 (b) $\underset{n\to \infty}{\mathrm{limsup}}}\phantom{\rule{0.2em}{0ex}}({a}_{n}+{b}_{n})\le {\displaystyle \underset{n\to \infty}{\mathrm{limsup}}}\phantom{\rule{0.2em}{0ex}}{a}_{n}+{\displaystyle \underset{n\to \infty}{\mathrm{limsup}}}\phantom{\rule{0.2em}{0ex}}{b}_{n$,
provided none of the sums is of the form $\infty \infty $.
Show by an example that strict inequality can hold.
 (c)
If ${a}_{n}\le {b}_{n}$ for all $n$ then
$\underset{n\to \infty}{\mathrm{liminf}}}\phantom{\rule{0.2em}{0ex}}{a}_{n}\le {\displaystyle \underset{n\to \infty}{\mathrm{liminf}}}\phantom{\rule{0.2em}{0ex}}{b}_{n$.

 (a)
Suppose
$f:X\to [\infty ,\infty ]$
and
$g:X\to [\infty ,\infty ]$
are measurable. Prove that the sets
$\left\{x\phantom{\rule{0.2em}{0ex}}\right\phantom{\rule{0.2em}{0ex}}f\left(x\right)<g\left(x\right)\left\}\phantom{\rule{2em}{0ex}}\text{and}\phantom{\rule{2em}{0ex}}\right\{x\phantom{\rule{0.2em}{0ex}}\left\phantom{\rule{0.2em}{0ex}}f\right(x)=g(x\left)\right\}$
 
are measurable.
 (b)
Prove that the set of points at which a sequence of measurable realvalued functions converges
(to a finite limit) is measurable.

Let $X$ be an uncountable set, let $\mathcal{M}$ be the collection
of all sets $E\subseteq X$ such that either $E$
or ${E}^{c}$ is at most countable,
and define $\mu \left(E\right)=0$ in the first case,
$\mu \left(E\right)=1$ in the second.
Prove that $\mathcal{M}$ is a $\sigma $algebra on $X$
and that $\mu $ is a measure on $\mathcal{M}$. Describe the
corresponding measurable functions and their integrals.
 Suppose ${f}_{n}:X\to [0,\infty ]$
is measurable for $n=1,2,3,\dots $,
${f}_{1}\ge {f}_{2}\ge {f}_{3}\ge \cdots \ge 0$,
${f}_{n}\left(x\right)\to f\left(x\right)$
as $n\to \infty $
for every $x\in X$, and
${f}_{1}\in {L}^{1}\left(\mu \right)$.
Prove that then
$\underset{n\to \infty}{\mathrm{lim}}{\int}_{X}{f}_{n}\phantom{\rule{0.1em}{0ex}}d\mu ={\int}_{X}f\phantom{\rule{0.1em}{0ex}}d\mu$
 
and show that this conclusion does not follow if the condition
"${f}_{1}\in {L}^{1}\left(\mu \right)$" is omitted.
 Put ${f}_{n}={\chi}_{E}$
if $n$ is odd, and
${f}_{n}=1{\chi}_{E}$
if $n$ is even. What is the relevance of this example to Fatou's lemma?
 Suppose $\mu $ is a positive measure on $X$,
$f:X\to [0,\infty ]$
is measurable, and
${\int}_{X}f\phantom{\rule{0.1em}{0ex}}d\mu =c$ where $0<c<\infty $
and $\alpha $ is a constant. Prove that
$\underset{n\to \infty}{\mathrm{lim}}{\int}_{X}n\phantom{\rule{0.1em}{0ex}}\mathrm{log}(1+{(f/n)}^{\alpha})\phantom{\rule{0.1em}{0ex}}d\mu =\{\begin{array}{ll}\infty ,& \text{if}\phantom{\rule{0.2em}{0ex}}0<\alpha <1,\\ c,& \text{if}\phantom{\rule{0.2em}{0ex}}\alpha =1,\\ 0,& \text{if}\phantom{\rule{0.2em}{0ex}}1<\alpha <\infty .\end{array}$
 
Hint: If $\alpha \le 1$, the integrands are dominated
by $\alpha f$. If $\alpha <1$, Fatou's
lemma can be applied.

Suppose $\mu \left(X\right)<\infty $,
${f}_{n}$ is a sequence of bounded complex measurable functions
on $X$, and ${f}_{n}\to f$
uniformly on $X$. Prove that
$\underset{n\to \infty}{\mathrm{lim}}{\int}_{X}{f}_{n}\phantom{\rule{0.1em}{0ex}}d\mu ={\int}_{X}f\phantom{\rule{0.1em}{0ex}}d\mu$.
 
 Show that
$A=\bigcap _{n=1}^{\infty}\bigcup _{k=n}^{\infty}{E}_{k}$
 
in [Ru, Theorem 1.41], and hence prove the theorem without any reference to integration.
 Suppose $f\in {L}^{1}\left(\mu \right)$. Prove that to each $\u03f5>0$ there exists
a $\delta >0$ such that
${\int}_{X}\leftf\right\phantom{\rule{0.1em}{0ex}}d\mu}<\u03f5$
whenever $\mu \left(E\right)<\delta $.
 Show that [Ru, Proposition 1.24(c)] is also true when $c=\infty $.
Notes and References
These exercises are taken from [Ru] for a course in "Measure Theory" at the Masters level at University of Melbourne.
References
[Ru]
W. Rudin,
Real and complex analysis, Third edition, McGrawHill, 1987.
MR??????.
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