Exercise 20, § 2

Let $F$ be a Fréchet space of countable type, $F'$ its dual, $\mu$ a positive measure on $T$, $m$ a measure on $T$ with values in the weak dual $F'_s$, scalarwise of basis $\mu$. For $m$ to be of basis $\mu$, it is necessary and sufficient that the following condition be fulfilled: for every $\varepsilon > 0$ and every compact subset $K$ of $T$, there exists a compact subset $K_1 \subset K$ such that $\mu(K - K_1) \leq \varepsilon$, and that the image by $m$ of the set of $\mu$-measurable functions $g$, bounded, with support contained in $K_1$ and verifying $\int |g| d\mu \leq 1$, is an equicontinuous subset of $F'$. (One will recall that $\int g d\mathbf{m} \in F'$ for every bounded $\mu$-measurable function $g$ with compact support; cf. cor. 2 of prop. 3. To show that the condition is necessary, use prop. 13 of § 1, No. 5 and prop. 5 of § 1, No. 2. To see that the condition is sufficient, consider first the case where $T$ is compact; applying cor. 3 of th. 1, No. 5, first show that there is a partition of $T$ into a negligible set $N$ and a sequence $(K_n)$ of compact sets, and a measurable mapping $g$ from $T$ into $F'_s$, such that $\int f d\mathbf{m} = \int g f d\mu$ for every function $f \in \mathscr{L}^\infty(\mu)$ null in the complement of a union of a finite number of sets $K_n$; to prove that $\mathbf{m} = g . \mu$, use exerc. 18 b) and 22 a) of chap. V, § 5. Finally, to pass to the case where $T$ is an arbitrary locally compact space, use prop. 4 of chap. V, § 1, No. 4.)