Exercise 12, § 2

a) Let K be a compact space, $E = E(K)$ the subspace of $\mathbf{R}^K$ formed by the linear combinations of characteristic functions of open sets; one identifies E with a subspace of the bidual of $\mathscr{C}(K)$. Show that a Cauchy sequence for $\sigma(\mathscr{M}(K), E)$ is convergent for $\sigma(\mathscr{M}(K), (\mathscr{M}(K))')$ (make use of Prop. 12 and Exer. 15 of Ch. V, §5). From this, deduce that every subset A of $\mathscr{M}(K)$ that is relatively compact for the topology $\sigma(\mathscr{M}(K), E)$, is relatively compact for the topology $\sigma(\mathscr{M}(K), (\mathscr{M}(K))')$. (Reduce to the case that K is metrizable, using Exer. 10 b). Show that A is bounded, by applying Exer. 13 of Ch. V, §5. Note that on A, the vague topology $\sigma(\mathscr{M}(K), \mathscr{C}(K))$ and the topology $\sigma(\mathscr{M}(K), \overline{E})$ are identical, $\overline{E} \supset \mathscr{C}(K)$ being the closure of E in the strong dual of $\mathscr{M}(K)$; deduce from this, using the fact that $\mathscr{C}(K)$ is separable, that every sequence of points of A has a subsequence that is Cauchy for $\sigma(\mathscr{M}(K), E)$; conclude by invoking Eberlein’s theorem.)

b) Let T be a locally compact space, m a vectorial measure on T with values in a quasi-complete, Hausdorff locally convex space F. Assume that, for every compact subset K of T, $\int_K dm \in F$; show that, for every bounded Borel f function on T with compact support, one has $\int f dm \in F$, and that the image under $f \mapsto \int f dm$ of the set of Borel functions with support contained in a compact subset K of T and of norm $\|f\| \leq 1$, is relatively compact for $\sigma(F, F')$ (make use of a) and Exer. 11, applied to $u : f \mapsto \int f dm$, where f runs over the set of linear combinations of characteristic functions of compact subsets of T).

c) Let F be a quasi-complete, Hausdorff locally convex space, m a vectorial measure on T with values in F, scalarly of base $\mu$; for every $z' \in F'$, write $z' \circ m = g_{z'} \cdot \mu$. For the condition of b) to be fulfilled, it is necessary and sufficient that, for every compact subset K of T, every equicontinuous subset H' of F', and every $\varepsilon > 0$, there exist a $\delta > 0$ such that the relations $A \subset K$ and $\mu^*(A) \leq \delta$ imply $\int_A^* |g_{z'}| d\mu \leq \varepsilon$ for every $z' \in H'$ (use Exer. 11 above and Exer. 15 of Ch. V, §5); m is then said to be absolutely continuous with respect to $\mu$ (for the original topology of F). Every vectorial measure with values in a semi-reflexive space and scalarly of base $\mu$, is absolutely continuous with respect to $\mu$ (Cor. 2 of Prop. 3). Every majorizable vectorial measure m with values in a Banach space $F$ is absolutely continuous with respect to $|\mathbf{m}|$ (for the original topology of $F$).