Proposition 7

— Let $f$ be a mapping of $T$ into $F$, scalarly essentially $\mu$-integrable, such that for every compact subset $K$ of $T$, $f(K)$ is contained

(*) Recall that for a bounded positive measure $\mu$, the concepts of $\mu$-integrable function and essentially $\mu$-integrable function are the same (Ch. V, §1, No. 3, Cor. of Prop. 9).

5 The original notation for upper essential integral is $\overline{\int^*}$, changed to $\int^\bullet$ in the second edition of Ch. V.

in a weakly compact, balanced, convex subset of F. Then $\int f d\mu$ belongs to the bidual $F''$ of F.

For every compact subset K of T,

the Cor. of Prop. 5 can be applied to the bounded measure $\varphi_K \cdot \mu$ and the function $f \varphi_K$, consequently $\int f \varphi_K d\mu \in F$. For every $z' \in F'$, $\langle z', f \rangle$ is essentially $\mu$-integrable, consequently (Ch. V, §1, No. 3, Prop. 10)

the limit being taken with respect to the increasing directed set of compact subsets of T. One concludes that, with respect to this set, $\int f \varphi_K d\mu$ converges to $\int f d\mu$ for the topology $\sigma({F'}^*, F')$. Now,

which proves that the set of elements $\int f \varphi_K d\mu$ is a bounded subset of $F_\sigma$, hence also of F (TVS, IV, §1, No. 1, Prop. 1). Proposition 7 is therefore a consequence of the following lemma: