CorollaryStatement: Literature-sourcedProof: AI-adaptedPipeline-generatedjudge pass (gpt-5.6-terra)audited 2026-09-07

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

Statement

Every Borel measure finite on compact sets on a second-countable LCH space is regular.

Facts & Assumptions

Given: $X$ 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]

If every open set is sigma-compact, compact-finite Borel measures are regular. (Sigma-compact open sets make locally finite Borel measures regular)

Proof

technique · direct

1.1

Refining a countable base by [L1] gives a countable base $(V_{n})$ with compact closures. Every open $U$ is the union of those $V_{n}$ whose closures lie in $U$ , hence is a countable union of compact sets $\overline{V_{n}}$ .

2.1

Thus every open set is sigma-compact, and [L2] applies to $μ$ .

step 1.1L2∎

Depends on

Used by

A locally integrable density functional is represented by g dlambda Example

Dependency tree · two levels

21 results within two dependency steps of this one, each drawn at its shortest distance from it. An arrow runs from a result to what uses it, so the chart reads left to right and ends at this result, which carries a heavier outline. Every node is a link to that result. Click elsewhere on the chart to enlarge it.

Sources

Donald L. Cohn, Measure Theory, 2nd ed., Chapter 7 (standard reference, not scraped)