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Locally finite Borel measures on second-countable LCH spaces are regular

Statement

Assume the Axiom of Countable Choice ${A C}_{ω}$ . Every Borel measure finite on compact sets on a second-countable LCH space is regular.

Facts & Assumptions

Given: Countable Choice (The Axiom of Countable Choice ( ${A C}_{ω}$ )), $X$ is second-countable and LCH, and $μ$ is finite on compact sets.

[L1]

An LCH space has a base of open sets with compact closure. (In a locally compact Hausdorff space every open set containing a point contains an open set containing it whose closure is compact and still inside; such a space is regular)

[L2]

Under Countable Choice, if every open set is sigma-compact, compact-finite Borel measures are regular (Sigma-compact open sets make locally finite Borel measures regular). The stated choice assumption is spent only in this theorem.

Proof

technique · direct

1.1L1givenconstruct

Let $(B_{n})$ be a given countable base and let $U$ be open. Take all indices $n$ for which $\overline{B_{n}}$ is compact and contained in $U$ ; this is a specified subfamily of the given base and needs no selection. For each $x \in U$ , [L1] gives an open $W$ with $x \in W \subseteq \overline{W} \subseteq U$ and $\overline{W}$ compact. Choose one $B_{n}$ with $x \in B_{n} \subseteq W$ ; its closure is a closed subset of the compact $\overline{W}$ , hence compact and contained in $U$ . Thus the selected subfamily covers $U$ , and $U$ is the union of the corresponding countably many compact closures.

2.1step 1.1L2given∎

Thus every open set is sigma-compact. Apply [L2] under the stated Countable Choice premise to obtain regularity of $μ$ .

Depends on

Dependency tree · 2 levels

Sources

Donald L. Cohn, Measure Theory, 2nd ed., Chapter 7