Exercise 6, § 2

Let F be a semi-reflexive locally convex space, equipped with an algebra structure over $\mathbf{R}$, such that the mapping $(u, v) \mapsto uv$ of $F \times F$ into F is separately continuous. Let m be a measure on T, with values in F, such that $m(fg) = m(f)m(g)$ for f and g in $\mathscr{H}(T)$. Show that if the numerical functions f, g are such that f, g, and fg belong to $\mathscr{L}(m)$, then $\int fg\, dm = (\int f\, dm)(\int g\, dm)$. (First treat the case that $g \in \mathscr{H}(T)$, by considering the two measures $f \mapsto m(fg)$ and $f \mapsto m(f)m(g)$; pass to the general case by proceeding similarly and making use of Exer. 5.) The case where $F = \mathscr{L}(G)$, G being reflexive and F equipped with the topology of pointwise convergence or the topology of weak pointwise convergence.