Lemma 4

Let F be a Hausdorff locally convex space, $f$ an element of $\mathscr{L}_{F'_s}^\infty$. For every numerical function $g \in \overline{\mathscr{L}}^1$, the function $gf$ is scalarly essentially $\mu$-integrable and $\int gf\, d\mu \in F'$.

For every $z \in F$, $\langle z, f \rangle$ belongs to $\mathscr{L}^\infty$, whence the first assertion. One can suppose moreover, without modifying $\int gf\, d\mu$, that $f(T) \subset V^\circ$, where V is a balanced convex neighborhood of 0 in F. Then the relation $z \in V$ implies $|\langle z, f(t) \rangle| \leq 1$ for all $t \in T$, whence $|\langle z, \int gf\, d\mu \rangle| = |\int \langle z, f \rangle g\, d\mu| \leq \overline{N}_1(g)$, which proves that $\int gf\, d\mu \in F'$.