Definition 1

Let $E$ be a locally convex space. One calls promeasure on $E$ every inverse system$^{(1)}$ of measures (§ 4, No. 2, Def. 1) on the inverse system of finite-dimensional quotients of $E$.

In other words, a promeasure $\mu$ on $E$ is a family $(\mu_V)_{V \in \mathscr{F}(E)}$, where $\mu_V$ is a bounded (positive) measure on the finite-dimensional space $E/V$, and where $\mu_V = p_{VW}(\mu_W)$ when $V \supset W$. All of the measures $\mu_V$ have the same total mass, which is called the total mass of the promeasure $\mu$.

For a subspace $V$ of $E$ to belong to $\mathscr{F}(E)$, it is necessary and sufficient that there exist a finite number of elements $x'_1, \ldots, x'_n$ of $E'$ such that $V$ consists of the $x \in E$ satisfying $\langle x, x'_i \rangle = 0$ for $1 \leq i \leq n$ (TVS, II, § 6, No. 3, Cor. 2 of Th. 1 and No. 5, Cor. 2 of Prop. 7). Moreover, on a finite-dimensional vector space there exists one and only one Hausdorff topological vector space topology (TVS, I, § 2, No. 3, Th. 2). Consequently, the concept of promeasure on $E$ depends only on the dual $E'$ of $E$.

Let $\lambda$ be a bounded measure on $E$. For every $V \in \mathscr{F}(E)$, let us denote by $\widetilde{\lambda}_V$ the image of $\lambda$ under the canonical mapping $p_V$ of $E$ onto $E/V$. One has $p_V = p_{VW} \circ p_W$ for any two elements $V$ and $W$ of $\mathscr{F}(E)$ such that $V \supset W$; consequently, the family $\widetilde{\lambda} = (\widetilde{\lambda}_V)_{V \in \mathscr{F}(E)}$ is a promeasure on $E$. We shall say that $\widetilde{\lambda}$ is the promeasure associated with the measure $\lambda$. One sees immediately that $\lambda$ and $\widetilde{\lambda}$ have the same total mass.