Exercise 23, § 1

Let $\mu$ be a positive measure on $T$, carried by a countable union of compact sets. a) Let $H$ be a set of $\mu$-measurable numerical functions, such that for every $t \in T$, $\sup_{f \in H} f(t) < +\infty$. Show that there exists a finite $\mu$-measurable function $g$ such that $f(t) \leq g(t)$ almost everywhere for every function $f \in H$. (Reduce to the case that $\mu$ is bounded and the functions in $H$ take their values in $[-1, +1]$, by means of an increasing homeomorphism of $\overline{\mathbf{R}}$ onto this interval. Then consider the supremum $\widetilde{g}$ in $L^1(\mu)$ of the set of classes of the functions in $H$ (Ch. IV, §3, No. 6, Prop. 14) and observe that there exists a sequence of functions in $H$ that converges almost everywhere to $g$.) b) Let $f$ be a scalarly $\mu$-measurable mapping of $T$ into a Hausdorff locally convex space $F$. For every subset $B'$ of $F'$, bounded for $\sigma(F', F)$, show that there exists a measurable and finite numerical function $g_{B'}$ such that, for every $z' \in B'$, $|\langle f(t), z' \rangle| \leq g_{B'}(t)$ almost everywhere. c) Let $F$ be a Fréchet space, $f$ a scalarly $\mu$-measurable mapping of $T$ into $F$. Show that for every compact subset $K$ of $T$ and every $\varepsilon > 0$, there exists a compact set $K' \subset K$ such that $\mu(K - K') \leq \varepsilon$ and the restriction of $f$ to $K'$ is bounded (make use of b)).