Exercise 3, § 2

Let m be a vectorial measure on T, with values in F. Show that for every numerical function $f$ essentially integrable for m, one has $\int f\, dm \in F''$ in each of the following two cases: a) F is distinguished (TVS, IV, §2, Exer. 4); b) F is a Fréchet space and T is countable at infinity. (In both cases, embed F in $F''$ and apply the method of Cor. 1 of Prop. 3; in case b), use the Cor. of Prop. 3 of TVS, IV, §3, No. 3.)