Proposition 18

On the cone $\mathscr{M}_+(X)$, the topologies induced by the following topologies coincide: $\mathscr{T}_1$: the topology of pointwise convergence in a linear subspace V of $\mathscr{K}(X; \mathbf{C})$ that is dense in $\mathscr{K}(X; \mathbf{C})$ and satisfies the property (P) (No. 7, Prop. 9);

\textit{\mathscr{T}_2: the vague topology;}

\textit{\mathscr{T}_3: the topology of strictly compact convergence.}

Since every filter is the intersection of the ultrafilters finer than it (GT, I, §6, No. 4, Prop. 7), it suffices to show that if $\mathfrak{U}$ is an ultrafilter on $\mathscr{M}_+(X)$ that converges to a measure $\mu_0$ for the topology $\mathscr{T}_1$, then it also converges to $\mu_0$ for $\mathscr{T}_3$. Let $K$ be a compact subset of $X$; by hypothesis, there exists a function $h \in V$ that is $\geqslant 0$ on $X$ and takes values $> 0$ on $K$; it follows that every function $f \in \mathscr{K}(X, K; \mathbf{C})$ may be written $f = gh$ with $g \in \mathscr{K}(X, K; \mathbf{C})$, and if $c = \inf_{x \in K} h(x) > 0$ then $\|g\| \leqslant c^{-1}\|f\|$. By hypothesis, there exists a set $H_0 \in \mathfrak{U}$ such that, for every measure $\mu \in H_0$,

Consequently, for every function $f \in \mathscr{K}(X; \mathbf{C})$,

for every measure $\mu \in H_0$; this proves that the set $H$ of measures $h \cdot \mu$, where $\mu$ runs over $H_0$, is \emph{vaguely bounded}. If $\mathfrak{U}_0$ is the ultrafilter induced by $\mathfrak{U}$ on $H_0$, the image of $\mathfrak{U}_0$ under the mapping $\mu \mapsto h \cdot \mu$ is the base of an ultrafilter $\mathfrak{F}$ on $H$, and since $H$ is relatively compact for the topology of strictly compact convergence (Prop. 17 and No. 9, Prop. 15), $\mathfrak{F}$ is convergent to a measure $\nu_0$ for this topology. In other words, for any $\varepsilon > 0$ and any compact subset $L$ of $\mathscr{K}(X, K; \mathbf{C})$, there exists a subset $N$ of $H_0$ belonging to $\mathfrak{U}$ such that, for every function $g \in L$ and every pair of measures $\mu, \mu'$ belonging to $N$, one has $|\langle g, h \cdot \mu \rangle - \langle g, h \cdot \mu' \rangle| \leqslant \varepsilon$, that is,

Now, we saw above that the mapping $g \mapsto gh$ is an \emph{automorphism} of the Banach space $\mathscr{K}(X, K; \mathbf{C})$. We have thus shown that $\mathfrak{U}$ is a \emph{Cauchy filter} on $\mathscr{M}_+(X)$ for the topology of strictly compact convergence. \emph{A fortiori}, it is a Cauchy filter for vague convergence, and Prop. 14 of No. 9 shows that it is vaguely convergent to a measure $\mu_1$; moreover, since $V$ is dense in $\mathscr{K}(X; \mathbf{C})$, the hypothesis implies that $\mu_1 = \mu_0$; finally, since $\mathfrak{U}$ is a Cauchy filter for the topology of strictly compact convergence, it also converges to $\mu_0$ for this topology (GT, X, §1, No. 5, Prop. 5).

Q.E.D.