Problem set 2: Due September 28

From Notes: Problems 6, 11, 12, 13, 14.

Problem 1   Show that the smallest $\sigma$-algebra containing the sets

$$(a,\infty]\subset[-\infty,\infty]$$

for all $a\in\mathbb{R},$ is what is called above the `Borel' $\sigma$-algebra on $[-\infty,\infty].$

Problem 2   Let $(X,\mathcal{M},\mu)$ be any measure space (so $\mu$ is a measure on the $\sigma$-algebra $\mathcal{M}$ of subsets of $X).$ Show that the set of equivalence classes of $\mu$-integrable functions on $X,$ with the equivalence relation

$$f_1\equiv f_2\Longleftrightarrow \mu(\{x\in X;f_1(x)\not=f_2(x)\})=0.$$

is a normed linear space with the usual linear structure and the norm given by

$$\Vert f\Vert=\int_X\vert f\vert d\mu.$$

Problem 3   Let $(X,\mathcal{M})$ be a set with a $\sigma$-algebra. Let $\mu : \mathcal{M} \to \mathbb{R}$ be a finite measure in the sense that $\mu ( \phi) =0$ and for any $\left\{ E_i \right\}^{\infty}_{i=1} \subset \mathcal{M}$ with $E_i \cap E_j = \phi$ for $i \neq j$,

$$\mu \left( \bigcup^{\infty}_{i=1} E_i \right) = \sum^{\infty}_{i=1} \mu (E_i)$$ (1)

with the series on the right always absolutely convergenct (i.e., this is part of the requirement on $\mu$). Define

$$\left\vert \mu \right\vert (E) = \sup \sum^{\infty}_{i=1} \left\vert \mu (E_i) \right\vert$$ (2)

for $E \in \mathcal{M}$, with the supremum over all measurable decompositions $E = \bigcup^{\infty}_{i=1} E_i$ with the $E_i$ disjoint. Show that $\left\vert \mu \right\vert$ is a finite, positive measure.

Hint 1. You must show that $\left\vert \mu \right\vert (E)= \sum^{\infty}_{i=1} \left\vert \mu \right\vert (A_i)$ if $\bigcup_i A_i =E$, $A_i \in \mathcal{M}$ being disjoint. Observe that if $A_j= \bigcup_{l} A_{j l}$ is a measurable decomposition of $A_j$ then together the $A_{j l}$ give a decomposition of $E$. Similarly, if $E=\bigcup_j E_j$ is any such decomposition of $E$ then $A_{j l} = A_j \cap E_{l}$ gives such a decomposition of $A_j$.

Hint 2. See [1] p. 117!

Problem 4 (Hahn Decomposition)  

With assumptions as in Problem 3:

1. Show that $\mu_+ = \frac{1}{2} ( \left\vert \mu \right\vert + \mu )$ and $\mu_-=\frac{1}{2} ( \left\vert \mu \right\vert - \mu )$ are positive measures, $\mu = \mu_+ - \mu_-$. Conclude that the definition of a measure in the notes based on (4.17) is the same as that in Problem 3.
2. Show that $\mu_\pm$ so constructed are orthogonal in the sense that there is a set $E \in \mathcal{M}$ such that $\mu_-(E)=0,$ $\mu_+(X\setminus E)=0.$

   Hint. Use the definition of $\vert\mu\vert$ to show that for any $F\in\mathcal{M}$ and any $\epsilon >0$ there is a subset $F'\in \mathcal{M},$ $F'\subset F$ such that $\mu_+(F')\ge\mu_+(F)-\epsilon$ and $\mu_-(F')\le\epsilon.$ Given $\delta>0$ apply this result repeatedly (say with $\epsilon=2^{-n}\delta)$ to find a decreasing sequence of sets $F_1=X,$ $F_n\in\mathcal{M},$ $F_{n+1}\subset F_n$ such that $\mu_+(F_n)\ge\mu_+(F_{n-1})-2^{-n}\delta$ and $\mu_-(F_n)\le 2^{-n}\delta.$ Conclude that $G=\bigcap_nF_n$ has $\mu_+(G)\ge\mu_+(X)-\delta$ and $\mu_-(G)=0.$ Now let $G_m$ be chosen this way with $\delta=1/m.$ Show that $E=\bigcup_m G_m$ is as required.

Problem 5  

Now suppose that $\mu$ is a finite, positive Radon measure on a locally compact metric space $X$ (meaning a finite positive Borel measure outer regular on Borel sets and inner regular on open sets). Show that $\mu$ is inner regular on all Borel sets and hence, given $\epsilon >0$ and $E \in \mathcal{B}(X)$ there exist sets $K \subset E \subset U$ with $K$ compact and $U$ open such that $\mu (K) \geq \mu (E) - \epsilon$, $\mu (E) \geq \mu (U) - \epsilon$.

Hint. First take $U$ open, then use its inner regularity to find $K$ with $K' \Subset U$ and $\mu (K') \geq \mu (U) - \epsilon / 2$. How big is $\mu (E \backslash K')$? Find $V \supset K' \backslash E$ with $V$ open and look at $K = K' \backslash V$.