The Nikodym Metric Space: The Vitali-Hahn-Saks Theorem and the Dunford-Pettis Theorem

For measurable sets $A$ and $B$ in a finite measure space ($X, \mathcal{M}, \mu$), show that

Let $\left\{ A_{n}\right\}$ be a sequence of measurable sets that converges to the measurable set $A_{0}$ with respect to the Nikodym metric. Show that $A_{0} \simeq \cup_{n = 1}^{\infty }\left[\cap_{k = n}^{\infty } A_{k}\right]$.

Let ($X, \mathcal{M}, \mu$) be a finite measure space and $\nu : \mathcal{M} \rightarrow [0, \infty )$ a finitely additive set-function with the property that for each $\epsilon > 0$, there is a $\delta > 0$ such that for a measurable set $E$, if $\mu (E) < \delta$, then $\nu (E) < \epsilon$. Show that $\nu$ is a measure on $\mathcal{M}$.

Let ($X, \mathcal{M}, \mu$) be a finite measure space. Show that the normed linear space $L^{1}(X, \mu )$ is separable if and only if the associated Nikodym metric space is separable.

Give an example of a decreasing sequence $\left\{ \mu_{n}\right\}$ of measures on $\mathcal{M}$ such that the set-function $\mu$ defined by $\mu (E) = \lim \mu_{n}(E)$ is not a measure.

Let $\left\{ \mu_{n}\right\}$ a sequence of measures on $\mathcal{M}$ such that for each $E \in \mathcal{M}, \mu_{n+1}(E) \geq \mu_{n}(E)$. For each $E \in \mathcal{M}$, define $\mu (E) = \lim \mu_{n}(E)$. Show that $\mu$ is a measure on $\mathcal{M}$ if $\mu (X) < \infty$.

Verify the assertion, in the proof of the Dunford-Pettis Theorem, that it suffices to consider non-negative functions.

Find a bounded sequence in $L^{1}([a, b], m)$, where $m$ is Lebesgue measure, which fails to have a weakly convergent subsequence.

Find a measure space ($X, \mathcal{M}, \mu$) for which every bounded sequence in $L^{1}(X, \mu )$ has a weakly convergent subsequence.

Let $(X, \mathcal{M}, \mu )$ be a finite measure space and $\left\{ \nu_{n}\right\}$ a sequence of finite measures of $\mathcal{M}$, each of which is absolutely continuous with respect to $\mu$. Prove that the following are equivalent: