Integration: Exercises

Arun Ram
Department of Mathematics and Statistics
University of Melbourne
Parkville, VIC 3010 Australia
aram@unimelb.edu.au

Last updates: 8 March 2011

Measure Theory Problem Set 2

1. Let $(X,\Sigma )$ be a measure space. Let $f,g:X\to ℝ$ be measurable. Let $\epsilon >0$. Show that the set $\{x\in X|f\left(x\right)+\epsilon <g\left(x\right)\}$ is in $\Sigma$.
2. Let $(X,\Sigma )$ be a measure space and let $\mu :\Sigma \to [0,\infty ]$ be a measure.

   (i)   For $h:X\to ℝ$ a simple function, give the definition of $\int hd\mu$.

   (ii)   Let $f:X\to {ℝ}_{\ge 0}$ be integrable. Give the definition of $\int hd\mu$.

   (iii)   Working directly from the definition of the integral show that if $g:X\to ℝ$ is defined by $g\left(x\right)=3f\left(x\right)$ then $\int gd\mu =3\int fd\mu$.

3. Let $K$ be a compact subset of $ℝ$. Let $f:K\to ℝ$ be continuous. Let $\epsilon >0$. Show that there exist $n\in {\mathbb{Z}}_{\ge 0}$ and $a_0,a_1,\dots ,a_n\in ℝ$ such that for all $x\in K$, $$\left|f\right(x)-(a_0+a_{1}x+a_2{x}^2+\cdots +a_n{x}^n\left)\right|<\epsilon .$$
4. Show that if $\mu ,\nu$ are Borel probability measures on a standard Borel space then we can find (positive) ${\mu }_0,{\mu }_1$ with $$\mu ={\mu }_0+{\mu }_1{\mu }_0\ll \nu ,{\mu }_1\perp \nu .$$
5. Let $m^{*}$ be the outer measure used to define Lebesgue measure $m$. Carefully define $m^{*}$ and Lebesgue measurable sets. Show that if $B\subseteq ℝ$ is Lebesgue measurable then there are Borel $A_1,A_2\subseteq ℝ$ with $A_1\subseteq B\subseteq A_2$ and $m(A_2\backslash A_1)=0$.
6. Let ${\Sigma }_0\subseteq {\Sigma }_1$ be two σ-algebras on a set $X$. Let $\mu$ be a σ-finite measure on $(X,{\Sigma }_1)$ and let $f:X\to ℝ$ be measurable with respect to ${\Sigma }_1$. Show that there is a function $g:X\to ℝ$ which is measurable with respect to ${\Sigma }_0$ such that on any $B\in {\Sigma }_0$ $${\displaystyle {\int }_{B}g\left(x\right)d\mu \left(x\right)={\int }_{B}f\left(x\right)d\mu \left(x\right)}$$
7. Let $Y$ be a standard Borel space and let $\mu$ be a finite atomless Borel measure on $Y$. Then for all $c\in ℝ$ with $0\le c\le \mu \left(Y\right)$, we can find $B\subseteq Y$ with $\mu \left(B\right)=c$.
8. Show that there exists a subset $V\subseteq [0,1]$ which is not Lebesgue measurable.
9. Let $K$ be a compact metric space. Let $A\subseteq K$ be Borel. Let ${\nu }_1,{\nu }_2$ be finite Borel measures on $K$ with the property that ${\nu }_1\left(B\right)=0$ for all Borel $B\subseteq A^c$, and ${\nu }_2\left(B\right)=0$ for all Borel $B\subseteq A$. Let $\mu$ be the signed measure ${\nu }_1-{\nu }_2$ (i.e. $\mu \left(B\right)={\nu }_1\left(B\right)-{\nu }_2\left(B\right)$). Show that ${\nu }_1\left(A\right)+{\nu }_2\left(A^c\right)$ equals the supremum of $$\left\{\right|\int fd\mu \left|\right|f:K\to [-1,1]\text{is continuous}\}.$$
10. Let $K=[0,1]$, the unit interval in its usual topology. Let $P\left(K\right)$ be the space of probability measures on $K$ equipped with the topology generated by the basic open sets $$\{\mu \in P(K\left)\right|\int f_{1}d\mu \in (a_1,b_1),\int f_{2}d\mu \in (a_2,b_2),\dots ,\int f_{n}d\mu \in (a_n,b_n)\},$$ for $n\in {\mathbb{Z}}_{>0}$, $f_1,\dots ,f_n$ continuous functions from $K$ to $ℝ$, and $a_1,b_1,\dots ,a_n,b_n\in ℝ$. Let $U$ be the open interval $(0,\frac{1}{2})$. Is $A=\{\mu \in P(K\left)\right|\mu \left(U\right)=1\}$ closed in this topology?
11. Let $(X,\Sigma )$ be a finite measure space. Let ${\left(f_n\right)}_{n\in {\mathbb{Z}}_{>0}}$ be a sequence of measurable functions which converge pointwise to $f$. Then for any $\epsilon >0$ there is $A\in \Sigma$ with $\mu (X\backslash A)<\epsilon$ and $\left(f_n\right)$ converging uniformly to $f$ on $A$.
12. Equip $\{H,T\}$ with the discrete topology and let $\prod _{{\mathbb{Z}}_{>0}}\{H,T\}$ be the collection of all functions $f:{\mathbb{Z}}_{>0}\to \{H,T\}$ equipped with the product topology. For $\overrightarrow{i}=(i_1,i_2,\dots i_N)$ a sequence of distinct elements of ${\mathbb{Z}}_{>0}$ and $\overrightarrow{S}=(S_1,S_2,\dots ,S_N)$ a sequence of elements of $\{H,T\}$ let $$A_{\overrightarrow{i},\overrightarrow{S}}=\{f\in \prod _{{\mathbb{Z}}_{>0}}|f\left(i_1\right)=S_1,f\left(i_2\right)=S_2,\dots ,f\left(i_N\right)=S_N\}.$$ Let ${\Sigma }_0$ be the collection of all finite unions of the sets of the form $A_{\overrightarrow{i},\overrightarrow{S}}$. Define ${\mu }_0:\left\{A_{\overrightarrow{i},\overrightarrow{S}}\right\}\to {ℝ}_{\ge 0}$ by ${\mu }_0\left(A_{\overrightarrow{i},\overrightarrow{S}}\right)=2^{-N}$. Show that ${\mu }_0$ extends to a function on ${\Sigma }_0$ which is σ-additive on its domain.

Notes and References

These exercises are taken from problems by G. Hjorth 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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