Proposition 2

— Let T be a countable set, and E the vector space of real functions on T, equipped with the topology of pointwise convergence. Every promeasure on E is a measure.

For every $t \in T$, let $\varepsilon_t$ be the linear form $f \mapsto f(t)$ on E. One knows (TVS, II, §6, No. 6, Cor. 2 of Prop. 8) that the family $(\varepsilon_t)_{t \in T}$ is a basis of the vector space $E'$. Denote by $\Phi$ the set of finite subsets of 'T', and for every $J \in \Phi$ let $E_J$ be the set of functions on T that are zero at every point of J. Let $F \in \mathscr{F}(E)$; since the orthogonal $F^\circ$ of F is a finite-dimensional subspace of $E'$, there exists a $J \in \Phi$ such that $F^\circ$ is contained in the linear subspace G of $E'$ generated by the $\varepsilon_t$ for $t \in J$. Since $F^\circ \subset G$, we have $E_J = G^\circ \subset F^{\circ\circ} = F$ and the countable family $(E_J)_{J \in \Phi}$ is cofinal in $\mathscr{F}(E)$. The proposition then follows from Th. 2 of §4, No. 3.