Lemma 7.3.4DE1E6EDC0B48

Proof. Let $\chi \ge 0$ be a continuous function with compact support, let $N$ be a compact neighborhood of $\operatorname{Supp}\chi $, and put $W=\operatorname{Int}N$. The function $f_1$ is upper semicontinuous, hence bounded above on $N$, and $f$ is bounded below on $N$; since $f\le f_ j\le f_1$, there is $c\ge 0$ with $|f_ j|\le c$ on $N$ for all $j$. The positive Radon measures $(f_{j_ k}+c)\mu _{j_ k}$ on $W$ converge weakly to $\nu +c\mu $. By proposition 2.6.1, $\nu +c\mu $ is a positive Radon measure on $W$, and the convergence is vague. Since also $\mu _ j\to \mu $ vaguely, we obtain $\int \psi f_{j_ k}\, \mathrm d\mu _{j_ k}\to \int \psi \, \mathrm d\nu $ for every continuous function $\psi $ with compact support in $W$.

Fix $l$. By lemma 3.4.4, there are bounded continuous functions $g_ m$ on $N$ decreasing to $f_ l$. For $j_ k\ge l$ we have $f_{j_ k}\le f_ l\le g_ m$ on $N$, so

Letting $k\to \infty $ gives $\int \chi \, \mathrm{d}\nu \le \int \chi g_ m\, \mathrm{d}\mu $. Since $-c\le f_ l\le g_ m\le g_1$ on $N$ and $\mu (N)<\infty $, bounded convergence as $m\to \infty $ gives $\int \chi \, \mathrm{d}\nu \le \int \chi f_ l\, \mathrm{d}\mu $, and bounded convergence as $l\to \infty $ gives $\int \chi \, \mathrm{d}\nu \le \int \chi f\, \mathrm{d}\mu $. Since $\chi \ge 0$ was arbitrary, $\nu \le f\mu $. $\square$