Exercise 13, § 2

Let $G$ be a Hausdorff barreled space, $F = G'$ its dual; if, when $F$ is equipped with a topology compatible with the duality between $F$ and $G$, $\mathbf{m}$ is a vectorial measure on $T$ with values in $F$, then $\mathbf{m}$ is again a vectorial measure when $F$ is equipped with the strong topology of the dual of $G$.

Assume $\mathbf{m}$ to be scalarly of base $\mu$; then $\mathbf{m}$ is absolutely continuous with respect to $\mu$, when $F$ is equipped with the topology $\tau(F, G)$ (Exer. 12 c)). In order that $\mathbf{m}$ be absolutely continuous with respect to $\mu$ when $F$ is equipped with the strong topology, it is necessary and sufficient that, for every compact subset $K$ of $T$ and every sequence $(A_n)$ of $\mu$-measurable subsets of $K$, the image $B$ under $\mathbf{m}$ of the closure in $\mathscr{L}^1(\mu)$ of the set of step functions of absolute value $\leq 1$, over the clan generated by the sequence of the $A_n$ (Ch. IV, §4, No. 9), contain a dense countable subset for the strong topology of $G'$. (To see that the condition is sufficient, reduce to the case that $T = K$ is a Stone space (Ch. IV, §4, Exer. 10). Then argue by contradiction: if $(A_n)$ is a sequence of subsets of $K$ that are both open and closed, one can assume, on passing to a metrizable quotient space $K_1$ of $K$ (Exer. 10 a)), that the continuous functions on $K_1$ corresponding to the $\varphi_{A_n}$ form a total set in $\mathscr{C}(K_1)$. On the other hand, on considering the quotient of $G$ by the subspace orthogonal to $B$, one can suppose that $B$ is total in $F = G'$ for the topology $\sigma(G', G)$; if $H$ is the subspace of $F = G'$ generated by $B$, the hypothesis then implies that every bounded subset $C$ of $G$ is precompact and metrizable for $\sigma(G, H)$. Therefore every bounded sequence $(z_n)$ of points of $G$ has a subsequence Cauchy for $\sigma(G, H)$; show that if $(z_n)$ is a Cauchy sequence for $\sigma(G, H)$, and if one writes $z_n \circ \mathbf{m} = g_{z_n} \cdot \mu$, the sequence of classes of the $g_{z_n}$ in $L^1(\mu)$ is a Cauchy sequence for the topology $\sigma(L^1, L^\infty)$; finally, conclude a contradiction from this by using Exers. 15 and 16 of Ch. V, §5.)