Second Problem set for 18.155, Fall 2002
Due September 24 in class or 2-174.

Note that the books of Folland [1] and Rudin [2] cover most of the material in the early lectures. These problems are concerned with the Hahn decomposition theorem for measures - I will assume this in proving the Riesz representation theorem.

Problem 1   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 [2] p. 117!

Problem 2 (Hahn Decomposition)  

With assumptions as in Problem 1:

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 signed measure as simply the difference of two positive measures is the same as that in Problem 1.
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 3  

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 $K = K' \backslash V$.

Problem 4   Using Problem 3 show that if $\mu$ is a finite Borel measure on a locally compact metric space $X$ then the following three conditions are equivalent

1. $\mu = \mu_1 - \mu_2$ with $\mu_1$ and $\mu_2$ both positive finite Radon measures.
2. $\left\vert \mu \right\vert$ is a finite positive Radon measure.
3. $\mu_+$ and $\mu_-$ are finite positive Radon measures.