PomeLemma 21.6.2.F2FABB0A312F Let $\mu _ j$ and $\mu $ be finite positive Radon measures on a finite-dimensional real vector space $W$ such that $\mu _ j\to \mu $ vaguely and $\mu _ j(W)\to \mu (W)$. Then $\int h\, \mathrm d\mu _ j\to \int h\, \mathrm d\mu $ for every bounded continuous function $h$ on $W$, that is, $\mu _ j\to \mu $ narrowly (section 2.6). In particular, if $P\subset M_{\mathbb R}$ is a convex body, $F_ j,F\in \mathcal E(N_{\mathbb R},P)$ and $F_ j\to F$ locally uniformly, then $\operatorname {MA}_{\mathbb R}(F_ j)\to \operatorname {MA}_{\mathbb R}(F)$ narrowly.
Lemma 21.6.2F2FABB0A312F
Proof. Let $\varepsilon >0$. By inner regularity there is a compact set $S$ with $\mu (W\setminus S)<\varepsilon $; let $\eta $ be a continuous function with compact support, $0\leq \eta \leq 1$ and $\eta =1$ on $S$. Since $\int \eta \, \mathrm d\mu _ j\to \int \eta \, \mathrm d\mu $ and the total masses converge, we have
\[ \lim _ j\int (1-\eta )\, \mathrm d\mu _ j =\int (1-\eta )\, \mathrm d\mu <\varepsilon , \]
so that $\mu _ j(W\setminus \operatorname{Supp}\eta )\leq \int (1-\eta )\, \mathrm d\mu _ j<\varepsilon $ for all large $j$. Thus the sequence $(\mu _ j)$ is eventually uniformly tight, and proposition 2.6.3 gives narrow convergence. (If $W$ is a point, every function on $W$ has compact support, and vague and narrow convergence coincide.) For the real Monge–Ampère measures, theorem 21.5.2 gives vague convergence and the common total mass $n!\operatorname {vol}(P)$. $\square$
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