2.8. Radon measures and convergence results [019T]

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2.8. Radon measures and convergence results

We shall make frequent use of basic integration and measure theory. Let $X$ be a compact (Hausdorff) space. A Radon measure on $X$ is a positive linear functional $\mu:C^{0}(X)\to\mathbf{R}$ . With this definition, it follows from the Riesz representation theorem that Radon measures are in 1-1 correspondence with regular Borel measures on $X$ ; see [Fol99, §7.1–2].

Since we shall be dealing with (possibly uncountable) nets rather than sequences, one has to be careful using results from integration theory. For example, the monotone convergence theorem is of course not true for general nets. However, as the next results show, integration of semicontinuous functions against Radon measures is often well behaved.

Lemma 2.23.

[Fol99, Proposition 7.12]. If $\mu$ is a positive Radon measure on $X$ and $(f_{j})_{j}$ a decreasing net of usc functions on $X$ , converging pointwise to a (usc) function $f$ , then $\lim_{j}\int f_{j}\mu=\int f\mu$ .

In particular, one has

Lemma 2.24.

[Fol99, Corollary 7.13]. If $\mu$ is a positive Radon measure on $X$ and $f$ is a usc function on $X$ , then

\int f\mu=\inf\left\{\int g\mu\mid f\leq g,\,g\in C^{0}(X)\right\}

Corollary 2.25.

Let $(f_{j})_{j}$ a decreasing net of usc functions on $X$ converging pointwise to a (usc) function $f$ , and $(\mu_{j})_{j}$ a net of positive Radon measures on $X$ converging weakly to a positive Radon measure $\mu$ . Then

\limsup_{j}\int f_{j}\mu_{j}\leq\int f\mu.

Proof.

Upon replacing $\mu_{j}$ with $(\int\mu_{j})^{-1}\mu_{j}$ we may assume that the $\mu_{j}$ ’s are probability measures. Fix any $\varepsilon>0$ . By Lemma 2.24 there exists a continuous function $g\geq f$ on $X$ such that $\int g\mu<\int f\mu+\varepsilon$ . By Dini’s lemma, we have $f_{j}<g+\varepsilon$ for all $j\gg 1$ , hence

\limsup_{j}\int f_{j}\mu_{j}\leq\limsup_{j}\int g\mu_{j}+\varepsilon=\int g\mu+\varepsilon\leq\int f\mu+2\varepsilon.

since $\int g\mu_{j}\to\int g\mu$ by the definition of weak convergence. The result follows. ∎