Exercise 27, § 2

Let $m$ be a vector measure on $T$ with values in a separated locally convex space $F$.

(a) Now prove that, if the support of $m$ is finite, $m = \sum_{i=1}^n c_i e_{a_i}$, where the $c_i \in F$.

(b) If F is a Banach space, and if m is continuous for the topology of compact convergence, now prove that m has a compact support.

(c) Give an example of a measure with values in $\mathbf{R}^N$, with non-compact support, and which is continuous for the topology of compact convergence.