Proposition 16

— Let K be a compact subset of X, L the vector space of measures $\mu$ on X with support contained in K. On L, the topologies induced by the topology $\mathscr{T}$ of compact convergence on $\mathscr{C}'(X; \mathbf{C})$ and the topology $\mathscr{T}'$ of strictly compact convergence on $\mathscr{M}(X; \mathbf{C})$ (Ch. III, §1, No. 10) coincide.

It is clear that on L, the topology induced by $\mathscr{T}$ is finer than the topology induced by $\mathscr{T}'$. Conversely, let H be a compact subset of $\mathscr{C}(X; \mathbf{C})$, h a function in $\mathscr{K}(X; \mathbf{C})$ equal to 1 on K. It is clear that the set H' of functions $fh$, where $f$ runs over H, is strictly compact in $\mathscr{K}(X; \mathbf{C})$, and, for every measure $\mu \in L$, $\mu(f) = \mu(fh)$ for every function $f \in H$, whence the conclusion.