Exercise 8, § 1

Let $F$ he a Hausdorff locally convex space, $f$ a scalarly locally integrable mapping of $T$ into $F$ such that, for every compact subset $K$ of $T$, $\int_K f\,d\mu \in F$.

a) Show that if $f$ is scalarly essentially integrable, and if $F$ is semi-reflexive (or if merely every Cauchy sequence for $\sigma(F,F')$ converges in $F$ for this topology, when $T$ is countable at infinity), then $\int gf\,d\mu \in F$ for every function $g \in \mathscr{L}^\infty$.

b) Show that if $F$ is quasi-complete and if, for every continuous semi-norm $q$ on $F$, one has $\int^* q(f(t))\,d\mu(t) < +\infty$, then $f$ is scalarly essentially integrable and $\int gf\,d\mu \in F$ for every function $g \in \mathscr{L}^\infty$.

(In both cases, consider the increasing directed set of compact subsets of $T$.)