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3.3.6 Measures on a compact metric boundaryE0A7F26D42BE

In this section we use the Riesz representation theorem for continuous linear functionals on $C(X)$, with $X$ compact metric: a bounded real functional is integration against a unique finite signed regular Borel measure, and its operator norm is the measure's total variation. This is an explicit measure-theory prerequisite, not an unproved potential-theoretic input.

Theorem 3.3.7 (Compact-metric measure selection).2C91786CB405 Let $X$ be a compact metric space, and let $(\mu _ j)$ be finite signed Borel measures with $\sup _ j\| \mu _ j\| \leq M<\infty $. There is a subsequence and a finite signed Borel measure $\mu $, with $\| \mu \| \leq M$, such that

\[ \int _ X f\, d\mu _{j_ k}\longrightarrow \int _ X f\, d\mu \qquad (f\in C(X)). \]

If all the $\mu _ j$ are positive, then $\mu $ is positive.

Proof. If $X$ is empty or $M=0$, every measure is zero and the assertion is immediate. First construct a countable dense subset of $C(X)$. For each positive integer $n$, choose a finite $1/n$-net $x_{n,1},\ldots ,x_{n,N_ n}$, and put

\[ p_{n,i}(x)=\frac{(2/n-\operatorname {dist}(x,x_{n,i}))^+}{\sum _{l=1}^{N_ n}(2/n-\operatorname {dist}(x,x_{n,l}))^+}. \]

The denominator is everywhere positive. These continuous functions form a partition of unity, and $p_{n,i}(x)>0$ implies $\operatorname {dist}(x,x_{n,i})<2/n$. For $f\in C(X)$, choose rational $q_{n,i}$ within $1/n$ of $f(x_{n,i})$. Then $\sum _ iq_{n,i}p_{n,i}$ converges uniformly to $f$ by uniform continuity. All rational combinations for all $n$ form a countable dense family.

Enumerate that family as $(g_ l)$. For every $l$, the sequence $\int g_ l\, d\mu _ j$ is bounded by $M\| g_ l\| _\infty $. Successive subsequences and diagonal extraction yield a subsequence along which all these integrals converge. If $f\in C(X)$ and $g_ l$ is uniformly close to $f$, then

\[ \left|\int f\, d\mu _{j_ k}-\int f\, d\mu _{j_{k'}}\right| \leq 2M\| f-g_ l\| _\infty + \left|\int g_ l\, d\mu _{j_ k}-\int g_ l\, d\mu _{j_{k'}}\right|. \]

Hence the integrals of every $f$ are Cauchy. Their limit $L(f)$ is a real linear functional with $|L(f)|\leq M\| f\| _\infty $. Riesz representation gives the required $\mu $ with $\| \mu \| \leq M$. If the original measures are positive, $L(f)\geq 0$ for $f\geq 0$; the positivity part of Riesz representation makes $\mu $ positive. $\square$

This form also applies to compactified boundaries, which need not be subsets of a fixed Euclidean space.

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