Integration: Exercises

1. Does there exist an infinite $\sigma$-algebra which has only countably many members?
2. Prove an analogue of [Ru, Theorem 1.8] for $n$ functions.
3. Prove that if $f$ is a real function on a measurable space $X$ such that $\left\{x\phantom{\rule{0.2em}{0ex}}|\phantom{\rule{0.2em}{0ex}}f\left(x\right)\ge r\right\}$ is measurable for every rational $r$, then $f$ is measurable.
4. Let $\left\{{a}_{n}\right\}$ and $\left\{{b}_{n}\right\}$ be sequences in $\left[-\infty ,\infty \right]$ and prove that
(a) $\underset{n\to \infty }{\mathrm{limsup}}\phantom{\rule{0.2em}{0ex}}\left(-{a}_{n}\right)=-\underset{n\to \infty }{\mathrm{liminf}}\phantom{\rule{0.2em}{0ex}}{a}_{n}$.
(b) $\underset{n\to \infty }{\mathrm{limsup}}\phantom{\rule{0.2em}{0ex}}\left({a}_{n}+{b}_{n}\right)\le \underset{n\to \infty }{\mathrm{limsup}}\phantom{\rule{0.2em}{0ex}}{a}_{n}+\underset{n\to \infty }{\mathrm{limsup}}\phantom{\rule{0.2em}{0ex}}{b}_{n}$, 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 $\underset{n\to \infty }{\mathrm{liminf}}\phantom{\rule{0.2em}{0ex}}{a}_{n}\le \underset{n\to \infty }{\mathrm{liminf}}\phantom{\rule{0.2em}{0ex}}{b}_{n}$.
5. (a)   Suppose $f:X\to \left[-\infty ,\infty \right]$ and $g:X\to \left[-\infty ,\infty \right]$ are measurable. Prove that the sets
 $\left\{x\phantom{\rule{0.2em}{0ex}}|\phantom{\rule{0.2em}{0ex}}f\left(x\right)
are measurable.
(b)   Prove that the set of points at which a sequence of measurable real-valued functions converges (to a finite limit) is measurable.
6. Let $X$ be an uncountable set, let $ℳ$ 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 $ℳ$ is a $\sigma$-algebra on $X$ and that $\mu$ is a measure on $ℳ$. Describe the corresponding measurable functions and their integrals.
7. Suppose ${f}_{n}:X\to \left[0,\infty \right]$ 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.
8. 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?
9. Suppose $\mu$ is a positive measure on $X$, $f:X\to \left[0,\infty \right]$ is measurable, and ${\int }_{X}f\phantom{\rule{0.1em}{0ex}}d\mu =c$ where $0 and $\alpha$ is a constant. Prove that
 $\underset{n\to \infty }{\mathrm{lim}}{\int }_{X}n\phantom{\rule{0.1em}{0ex}}\mathrm{log}\left(1+{\left(f/n\right)}^{\alpha }\right)\phantom{\rule{0.1em}{0ex}}d\mu =\left\{\begin{array}{ll}\infty ,& \text{if}\phantom{\rule{0.2em}{0ex}}0<\alpha <1,\\ c,& \text{if}\phantom{\rule{0.2em}{0ex}}\alpha =1,\\ 0,& \text{if}\phantom{\rule{0.2em}{0ex}}1<\alpha <\infty .\end{array}$
Hint: If $\alpha \le 1$, the integrands are dominated by $\alpha f$. If $\alpha <1$, Fatou's lemma can be applied.
10. 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
 $\underset{n\to \infty }{\mathrm{lim}}{\int }_{X}{f}_{n}\phantom{\rule{0.1em}{0ex}}d\mu ={\int }_{X}f\phantom{\rule{0.1em}{0ex}}d\mu$.
11. 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.
12. Suppose $f\in {L}^{1}\left(\mu \right)$. Prove that to each $ϵ>0$ there exists a $\delta >0$ such that ${\int }_{X}|f|\phantom{\rule{0.1em}{0ex}}d\mu <ϵ$ whenever $\mu \left(E\right)<\delta$.
13. 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, McGraw-Hill, 1987. MR??????.