## Integration: Exercises

Last updates: 8 March 2011

## Measure Theory Problem Set 2

1. Let $\left(X,\Sigma \right)$ be a measure space. Let $f,g:X\to ℝ$ be measurable. Let $\epsilon >0$. Show that the set $\left\{x\in X\phantom{\rule{0.2em}{0ex}}|\phantom{\rule{0.2em}{0ex}}f\left(x\right)+\epsilon is in $\Sigma$.
2. Let $\left(X,\Sigma \right)$ be a measure space and let $\mu :\Sigma \to \left[0,\infty \right]$ be a measure.
(i)   For $h:X\to ℝ$ a simple function, give the definition of $\int h\phantom{\rule{0.1em}{0ex}}d\mu$.
(ii)   Let $f:X\to {ℝ}_{\ge 0}$ be integrable. Give the definition of $\int h\phantom{\rule{0.1em}{0ex}}d\mu$.
(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 $\int g\phantom{\rule{0.1em}{0ex}}d\mu =3\int f\phantom{\rule{0.1em}{0ex}}d\mu$.
3. Let $K$ be a compact subset of $ℝ$. Let $f:K\to ℝ$ be continuous. Let $\epsilon >0$. Show that there exist $n\in {ℤ}_{\ge 0}$ and ${a}_{0},{a}_{1},\dots ,{a}_{n}\in ℝ$ such that for all $x\in K$, $| f(x)-( a0+ a1x+ a2x2+ ⋯ + anxn) | <ε.$
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 $μ=μ0 +μ1 μ0≪ν, μ1⊥ν.$
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 $m\left({A}_{2}\{A}_{1}\right)=0$.
6. Let ${\Sigma }_{0}\subseteq {\Sigma }_{1}$ be two σ-algebras on a set $X$. Let $\mu$ be a σ-finite measure on $\left(X,{\Sigma }_{1}\right)$ 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 $B\in {\Sigma }_{0}$ $∫B g(x) dμ(x) = ∫B f(x) dμ(x)$
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 $V\subseteq \left[0,1\right]$ 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 ${|∫fdμ| | f:K→[-1,1] is continuous}.$
10. Let $K=\left[0,1\right]$, 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 ${ μ∈P(K) | ∫f1dμ ∈(a1,b1) , ∫f2dμ ∈(a2,b2) , … , ∫fndμ ∈(an,bn) },$ for $n\in {ℤ}_{>0}$, ${f}_{1},\dots ,{f}_{n}$ continuous functions from $K$ to $ℝ$, and ${a}_{1},{b}_{1},\dots ,{a}_{n},{b}_{n}\in ℝ$. Let $U$ be the open interval $\left(0,\frac{1}{2}\right)$. Is $A=\left\{\mu \in P\left(K\right)\phantom{\rule{.5em}{0ex}}|\phantom{\rule{.5em}{0ex}}\mu \left(U\right)=1\right\}$ closed in this topology?
11. Let $\left(X,\Sigma \right)$ be a finite measure space. Let ${\left({f}_{n}\right)}_{n\in {ℤ}_{>0}}$ be a sequence of measurable functions which converge pointwise to $f$. Then for any $\epsilon >0$ there is $A\in \Sigma$ with $\mu \left(X\A\right)<\epsilon$ and $\left({f}_{n}\right)$ converging uniformly to $f$ on $A$.
12. Equip $\left\{H,T\right\}$ with the discrete topology and let $\prod _{{ℤ}_{>0}}\left\{H,T\right\}$ be the collection of all functions $f:{ℤ}_{>0}\to \left\{H,T\right\}$ equipped with the product topology. For $\stackrel{\to }{i}=\left({i}_{1},{i}_{2},\dots {i}_{N}\right)$ a sequence of distinct elements of ${ℤ}_{>0}$ and $\stackrel{\to }{S}=\left({S}_{1},{S}_{2},\dots ,{S}_{N}\right)$ a sequence of elements of $\left\{H,T\right\}$ let $A i→, S→ ={ f∈ ∏ ℤ>0 | f(i1)=S1, f(i2)=S2, …, f(iN)=SN }.$ Let ${\Sigma }_{0}$ be the collection of all finite unions of the sets of the form ${A}_{\stackrel{\to }{i},\stackrel{\to }{S}}$. Define ${\mu }_{0}:\left\{{A}_{\stackrel{\to }{i},\stackrel{\to }{S}}\right\}\to {ℝ}_{\ge 0}$ by ${\mu }_{0}\left({A}_{\stackrel{\to }{i},\stackrel{\to }{S}}\right)={2}^{-N}$. 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??????.