Proposition 9

— Let T be a locally compact space, and let $\mathfrak{F}$ be a filter on $\mathscr{M}^b_+(T)$ that converges vaguely to a bounded measure $\mu$. For $\mathfrak{F}$ to converge tightly to $\mu$, it is necessary and sufficient that $\lim_{\lambda} \lambda(1) = \mu(1)$ with respect to $\mathfrak{F}$.

The condition is obviously necessary. To show that it is sufficient, let us denote by $X$ the Alexandroff compactification of $T$ (GT, I, §9, No. 8) and by $i$ the canonical injection of $T$ into $X$. By Prop. 8, everything comes down to showing that $\lambda \mapsto i(\lambda)$ converges tightly to $i(\mu)$ in $\mathscr{M}^b(X)$ with respect to $\mathfrak{F}$. Since $\mu(1) < +\infty$, there exists a set $A \in \mathfrak{F}$ such that the total masses of the measures in $A$ are bounded by a number $M$; it therefore suffices to verify that

for functions $g \in \mathscr{C}^b(X)$ forming a total set in $\mathscr{C}^b(X)$. Now, this equality is satisfied when $g$ has compact support in $T$, because of the vague convergence of $\mathfrak{F}$ to $\mu$, and also when $g$ is a constant function on $X$, from the fact that $\lim_{\lambda, \mathfrak{F}} \lambda(1) = \mu(1)$. Since the functions of the preceding two types form a total set in $\mathscr{C}^b(X)$ (Ch. III, §1, No. 2, Prop. 3), this completes the proof.