Proposition 6

The set of measures on a locally compact space $X$, whose support is contained in a closed set $F$, is a vaguely closed linear subspace of $\mathscr{M}(X; \mathbf{C})$.

For, it is the intersection of the vaguely closed hyperplanes with equation $\mu(f) = 0$, where $f$ runs over the set of functions in $\mathscr{H}(X; \mathbf{C})$ whose support does not intersect $F$.

Suppose $X$ is not compact: given a filter $\Phi$ on the space $\mathscr{M}(X; \mathbf{C})$ of measures on $X$, we shall say that the support of a measure $\mu$ recedes indefinitely along $\Phi$ if, for every compact subset $K$ of $X$, there exists a set $M \in \Phi$ such that $\mathrm{Supp}(\mu) \cap K = \varnothing$ for every measure $\mu \in M$.