Corollary 1

If, for every compact subset K of T, the image under m of the set of $f \in \mathscr{H}(T, K)$ such that $\sup_{t \in T} |f(t)| \leq 1$ is weakly relatively compact in F, then $\int f dm \in F$ for every function $f \in \mathscr{L}(m)$ that is bounded and has compact support, and $\int f dm \in F''$ for every function $f \in \mathscr{L}(m)$.

The first assertion may be deduced immediately from Prop. 3: if $f$ is bounded and has compact support, and if U is a relatively compact open neighborhood of the support of $f$, then the restriction of $m$ to the subspace $\mathscr{H}(U)$ is a measure $m_U$ on U that satisfies the conditions of Prop. 3, and $\int f dm_U = \int f dm$ (Ch. V, §7, No. 1, Th. 1), therefore $\int f dm \in F$.

Now let $f$ be any element of $\mathscr{L}(m)$; for every compact subset K of T and every integer $n > 0$, let $f_{n,K}$ be the numerical function on T defined as follows: if $t \notin K$, $f_{n,K}(t) = 0$; if $t \in K$ and $|f(t)| \leq n$, $f_{n,K}(t) = f(t)$; finally, if $t \in K$ and $|f(t)| > n$, $f_{n,K}(t) = nf(t)/|f(t)|$. It is clear that for every $t \in T$, $f(t)$ is the limit of $f_{n,K}(t)$ with respect to the product filter of the Fréchet filter by the section filter of the (increasing directed) ordered set of compact subsets of T. Since $|f_{n,K}| \leq |f|$, it follows from Lebesgue’s theorem and Prop. 10 of Ch. V, §1, No. 3, applied to each scalar measure $|z'\circ m|$, that $f_{n,K}$ converges to $f$ in $\mathscr{L}(m)$ with respect to the preceding filter. Consequently, the integral $\int f dm$ is in the closure in ${F'}^*$ (for the topology $\sigma({F'}^*, F')$) of the set M of $m(f_{n,K})$. But the first part of the corollary shows that $M \subset F$, and, on the other hand, for every $z' \in F'$ one has $|\langle z', m(f_{n,K})\rangle| \leq \int |f| d|z'\circ m|$, which shows that M is bounded in $F_\sigma$, hence also in F (TVS, IV, §1, No. 1, Prop. 1). Lemma 1 of §1, No. 2 therefore shows that $\int f dm \in F''$.