Exercise 20, § 2

Let $F$ be a separable Fréchet space, $F'$ its dual, $\mu$ a positive measure on $T$, $m$ a measure on $T$ with values in the weak dual $F'_s$, scalarly of base $\mu$. For $m$ to

³ According to the Feuille d’errata No 14, this exercise is to be suppressed.

have base $\mu$, 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 such that the image under $m$ of the set of $\mu$-measurable functions $g$ that are bounded, have support contained in $K_1$ and satisfy $\int |g| d\mu \leq 1$, is an equicontinuous subset of $F'$. (Recall that $\int g dm \in F'$ for every $\mu$-measurable function $g$ that is bounded and has 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 that $T$ is compact; applying Cor. 3 of Th. 1, No. 5, first show that there exist a partition of $T$ into a negligible set $N$ and a sequence $(K_n)$ of compact sets, and a measurable mapping $g$ of $T$ into $F'_s$, such that $\int f dm = \int gf d\mu$ for every function $f \in \mathscr{L}^\infty(\mu)$ that is zero on the complement of the union of a finite number of the sets $K_n$; to prove that $m = g \cdot \mu$, use Exers. 16 b) and 20 a) of Ch. V, §5. Finally, to pass to the case that $T$ is any locally compact space, use Prop. 14 of Ch. IV, §5, No. 9.)