Corollary

— The condition b) of Prop. 9 is satisfied in each of the following two cases:

$1^\circ$ The measure $\mu$ is bounded, and if $S$ is its support, then $U(S)$ is an equicontinuous subset of $\mathscr{L}(G; H)$.

$2^\circ$ The condition a) of Prop. 9 is satisfied, the space $G$ is barreled, and, for every compact subset $K$ of $T$, $U(K)$ is a bounded subset of $\mathscr{L}_s(G; H)$.

Let us first place ourselves in case $1^\circ$. We can restrict ourselves to the case that $S = T$ (Ch. V, §7, No. 1, Th. 1). Then, for every equicontinuous subset $B'$ of $H'$, there exists an equicontinuous, convex, balanced and weakly closed subset $A' \subset G'$ such that ${}^t U(t) \cdot \mathbf{y}' \in A'$ for all $\mathbf{y}' \in B'$ and all $t \in T$ (TVS, II, §6, No. 4, Prop. 6). Since $U$ is scalarly $\mu$-integrable, the mapping $t \mapsto {}^t U(t) \cdot \mathbf{y}'$ of $T$ into the dual $G'$ of $G$ equipped with $\sigma(G', G)$, is scalarly $\mu$-integrable, and one may write

Since $A'$ is convex and compact for $\sigma(G', G)$, the Cor. of Prop. 5 of No. 2 shows that $u_{\mathbf{y}'} \in \mu(T)A'$ for every $\mathbf{y}' \in B'$, which proves our assertion.

Let us now place ourselves in case $2^\circ$. For every $\mathbf{y}' \in H'$ and every compact subset $K$ of $T$, set

an element of the algebraic dual $G^*$ of $G$. Since $G$ is barreled, every bounded subset of $\mathscr{L}_s(G; H)$ is equicontinuous (TVS, III, §4, No. 2, Th. 1); the first part of the argument, applied to the function $\varphi_K U$ and the bounded measure $\varphi_K \cdot \mu$, shows that $u_{K, y'} \in G'$. Moreover, for the topology $\sigma(G^*, G)$, one has $u_{y'} = \lim_{K} = u_{K, y'}$, the limit being taken with respect to the increasing directed set of compact subsets of $T$ (Ch. V, §1, No. 3, Prop. 10). To verify condition b) of Prop. 9, it suffices, by Prop. 9, to prove that the linear mapping $S$ of $G$ into $H$ defined by (1) is continuous; moreover, $G$ is barreled, therefore it will suffice to prove that $S$ is continuous when $G$ and $H$ are equipped with their weakened topologies (TVS, IV, §1, No. 3, Prop. 7); finally, by virtue of (2), one is reduced to showing that $u_{y'} \in G'$ for every $y' \in H'$. Since $u_{y'}$ is in the closure, for $\sigma(G^*, G)$, of the set $M'$ of the $u_{K, y'}$, where $K$ runs over the set of compact subsets of $T$, it suffices to prove that $M'$ is equicontinuous (TVS, III, §3, No. 4, Prop. 4); and since $G$ is barreled, it comes to the same to say that for every $x \in G$, the set of $\langle x, u_{K, y'} \rangle$ is bounded (TVS, III, §4, No. 2, Th. 1). But this follows at once from the relations