Corollary 4

Let $F$ be a separable normed space, and $m$ a measure on $T$ with values in the strong dual $F'$, majorizable for the norm of $F'$. Then $m$ is a measure with base $|m|$, and if $m = f \cdot |m|$ then $|f(t)| = 1$ locally almost everywhere for $|m|$.

By hypothesis, for every $z \in F$ such that $|z| \leq 1$, one has $|\langle z, m(g) \rangle| \leq |m|(|g|)$ for every function $g \in \mathscr{K}(T)$, consequently $|m(g)| \leq |m|(|g|)$ (TVS, IV, §2, No. 4, formula (1)). Since every ball in $F'$ is equicontinuous, Cor. 3 is applicable and shows that $m$ has base $|m|$; moreover, if $m = f \cdot |m|$ then $f$ is $|m|$-measurable for $\sigma(F', F)$ (§1, No. 5, Prop. 13) and $|m| = |f| \cdot |m|$ (No. 4, Prop. 8), which proves the corollary (Ch. V, §5, No. 3, Cor. 2 of Prop. 3).

If this corollary is applied to the case that $F$ if finite-dimensional, one recovers as a special case the first part of Prop. 9.