Exercise 8, § 2

Let m be a vectorial measure on T with values in F, q-majorizable for a lower semi-continuous semi-norm q on F. Show that for every function $f \geq 0$ of $\mathscr{H}(T)$,

where $(f_i)$ runs over the set of all finite sequences of elements of $\mathscr{H}(T)$ such that $\sum_i |f_i| \leq f$. (Argue as in Th. 1 of Ch. II, §2, No. 2.)