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

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 $\left\{x\right|f\left(x\right)\ge r\}$ is measurable for every rational $r$, then $f$ is measurable.
4. Let $\left\{a_n\right\}$ and $\left\{b_n\right\}$ be sequences in $[-\infty ,\infty ]$ and prove that

   (a) ${\displaystyle \underset{n\to \infty }{\mathrm{limsup}}}(-a_n)=-{\displaystyle \underset{n\to \infty }{\mathrm{liminf}}}{a}_n$.

   (b) ${\displaystyle \underset{n\to \infty }{\mathrm{limsup}}}(a_n+b_n)\le {\displaystyle \underset{n\to \infty }{\mathrm{limsup}}}{a}_n+{\displaystyle \underset{n\to \infty }{\mathrm{limsup}}}{b}_n$, 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 ${\displaystyle \underset{n\to \infty }{\mathrm{liminf}}}{a}_n\le {\displaystyle \underset{n\to \infty }{\mathrm{liminf}}}{b}_n$.

5. (a)   Suppose $f:X\to [-\infty ,\infty ]$ and $g:X\to [-\infty ,\infty ]$ are measurable. Prove that the sets

   $\left\{x\right|f\left(x\right)<g\left(x\right)\left\}\text{and}\right\{x\left|f\right(x)=g(x\left)\right\}$

   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 $f_n:X\to [0,\infty ]$ 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

   ${\displaystyle \underset{n\to \infty }{\lim }{\int }_X{f}_{n}d\mu ={\int }_{X}fd\mu }$

   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 $X$, $f:X\to [0,\infty ]$ is measurable, and ${\int }_{X}fd\mu =c$ where $0<c<\infty$ and $\alpha$ is a constant. Prove that

   ${\displaystyle \underset{n\to \infty }{\lim }{\int }_{X}n\log (1+{(f/n)}^{\alpha })d\mu =\{\begin{array}{ll}\infty ,& \text{if}0<\alpha <1,\\ c,& \text{if}\alpha =1,\\ 0,& \text{if}1<\alpha <\infty .\end{array}}$

   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

    ${\displaystyle \underset{n\to \infty }{\lim }{\int }_X{f}_{n}d\mu ={\int }_{X}fd\mu }$.

11. Show that

    ${\displaystyle A=\bigcap _{n=1}^{\infty }\bigcup _{k=n}^{\infty }{E}_k}$

    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 ${\displaystyle {\int }_X\left|f\right|d\mu }<ϵ$ 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??????.

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