Exercise 5, § 2

Let F be a semi-reflexive locally convex space, m a measure on T with values in F. Show that for every numerical function g locally integrable for m, the mapping $f \mapsto \int fg\, dm$ of $\mathscr{H}(T)$ into F is a measure on T, denoted $g \cdot m$; in order that a numerical function h be essentially integrable for $g \cdot m$, it is necessary and sufficient that $gh$ be essentially integrable for m, in which case $\int h\, d(g \cdot m) = \int hg\, dm$.