Proposition 1

*Let $E$ be a locally convex space. The mapping $\lambda \mapsto \widetilde{\lambda}$ is a bijection of the set of bounded measures on $E$ onto the set of promeasures $(\mu_V)_{V \in \mathscr{F}(E)}$ on $E$ satisfying the following condition:

For every $\varepsilon > 0$, there exists a compact subset $K$ of $E$ such that $\mu_V(E/V - p_V(K)) \leq \varepsilon$ for all $V \in \mathscr{F}(E)$.*

One knows that the intersection of the kernels of the continuous linear forms on $E$ is equal to 0 (TVS, II, § 4, No. 1, Cor. 1 of Prop. 2); consequently $\bigcap_{V \in \mathscr{F}(E)} V = \{0\}$ and the family $(p_V)_{V \in \mathscr{F}(E)}$ is coherent and separating. The proposition then follows from Th. 1 of § 4, No. 2.

In particular, the mapping $\lambda \mapsto \widetilde{\lambda}$ is injective. If $\mu$ is a promeasure on $E$, and if there exists a bounded measure $\lambda$ on $E$ such that $\mu = \widetilde{\lambda}$, we shall say, by an abuse of language, that $\mu$ is a measure. If $E$ is finite-dimensional, every promeasure $\mu = (\mu_V)_{V \in \mathscr{F}(E)}$ is a measure: for, $\{0\} \in \mathscr{F}(E)$, $E/\{0\} = E$ and $p_{V,\{0\}} = p_V$, whence $\mu_V = p_V(\mu_{\{0\}})$ for all $V \in \mathscr{F}(E)$; in other words, $\mu = \widetilde{\lambda}$ with $\lambda = \mu_{\{0\}}$.

(1) Also called a 'projective system'.