Proposition 8

Let $\mu$ be a bounded positive measure on T, S a $\mu$-measurable set carrying $\mu$, f a $\mu$-measurable mapping of T into F, such that $f(S)$ is contained in a complete, bounded, balanced convex subset B of F. Then, f is scalarly $\mu$-integrable and $\int f d\mu \in \mu(T)B \subset F$.

Since S is $\mu$-integrable, there exists a partition of S formed by a $\mu$-negligible set N and a sequence $(K_n)$ of compact subsets such that the restriction of f to each $K_n$ is continuous (Ch. IV, §4, No. 6, Cor. 3 of Th. 4 and §5, No. 1, Def. 1); $f(K_n)$ is therefore a compact subset of $F$. The closed, balanced convex envelope $B_n$ of $f(K_n)$ is then pre-compact (TVS, II, §4, No. 1, Prop. 3) and is contained in the complete subset $B$ of $F$, therefore it is compact, and a fortiori weakly compact. Consequently (Cor. of Prop. 5) $f \varphi_{K_n}$ is scalarly $\mu$-integrable, and

For every continuous semi-norm $p$ on $F$, it follows that

since $B$ is bounded and since the series with general term $\mu(K_n)$ is convergent and has sum $\mu(T)$, one sees that the sequence with general term $s_n = z_1 + z_2 + \cdots + z_n$ is a Cauchy sequence in the complete subset $\mu(T)B$ of $F$. This sequence therefore converges to an element $s$ of $\mu(T)B$; since one can suppose that $f(t) = 0$ on $T - S$, Lebesgue’s theorem applied to each of the functions $\langle z', f \rangle$ ($z' \in F'$) proves that $s = \int f d\mu$.