Proposition 6

— *Let F be a quasi-complete, Hausdorff locally convex space, m a majorizable measure on T with values in F. If f is a numerical function essentially integrable for all of the measures q(m) (where q runs over the set of continuous semi-norms on F), then f is essentially integrable for m, and $\int f dm \in F$.

We shall make use of the following auxiliary result. Let $(\mu_\iota)_{\iota \in I}$ be an increasing directed family of positive measures on T. Let us denote by $\mathscr{L}^1((\mu_\iota)_{\iota \in I})$ the vector space of finite numerical functions on T, essentially $\mu_\iota$-integrable for every $\iota \in I$, equipped with the topology defined by the semi-norms $f \mapsto \mu_\iota(|f|)$ ($\iota \in I$). Let $\mathscr{L}_0$ be the linear subspace of $\mathscr{L}^1((\mu_\iota)_{\iota \in I})$ generated by the products $g \varphi_K$, where g runs over the set of continuous finite numerical functions on T, and K over the set of compact subsets of T.