Corollary 3

Let $F$ be a Hausdorff locally convex space containing a countable dense subset, $m$ a vectorial measure on $T$ with values in the weak dual $F'$ of $F$. If the image under $m$ of the set $B$ of functions $g$ in $\mathscr{K}(T)$ such that $\mu(|g|) \leq 1$ is contained in a closed and convex equicontinuous subset $H'$ of $F'$, then $m$ has base $\mu$ and there exists a density $f$ of $m$ with respect to $\mu$ such that $f(t) \in H'$ for all $t \in T$.

The hypothesis implies that $m$ is continuous when $\mathscr{K}(T)$ is equipped with the topology induced by that of $\mathscr{L}^1(\mu)$ (defined by the semi-norm $N_1$); it may therefore be extended to a continuous linear mapping $u$ of $\mathscr{L}^1(\mu)$ into the completion $G$ of the weak dual of $F$; but since $H'$ is a compact subset of $G$ and the image under $u$ of the set $\overline{B}$ of $f \in \mathscr{L}^1$ such that $N_1(f) \leq 1$ is contained in the closure of $H'$ in $G$, one has $u(\overline{B}) \subset H'$, therefore $u$ maps $\mathscr{L}^1$ into $F'$. Since the relation $N_1(f) \leq \varepsilon$ implies that $u(f) \in \varepsilon H'$, one has $u(g) = 0$ if $g$ is $\mu$-negligible, and Cor. 2 may therefore be applied to the mapping of $L^1$ into $F'$ obtained from $u$ by passing to the quotient; whence the corollary.