Exercise 9, § 2

Let T be a locally compact space countable at infinity, F a Hausdorff barreled space containing a countable dense subset, m a vectorial measure on T with values in the weak dual $F'$ of F. Show that there exists a positive measure $\mu$ on T such that m is scalarly of base $\mu$. (Make use of Prop. 12 of Ch. V, §5, No. 6, and condition 5) of Cor. 5 of the Lebesgue–Nikodym theorem, Ch. V, §5, No. 5, Th. 2.)