False statementConstruction: Literature-sourcedVerification: AI-adaptedPipeline-generatedjudge pass (gpt-5.6-terra)audited 2026-09-07

How statement and proof provenance work

The first chip identifies the source of the statement or construction; the second identifies the source of its local proof or verification.

Literature-sourced: the exact statement appears in a cited source; only wording and notation differ.
AI-adapted: a semantically identical restatement of literature-sourced material, modulo indexing, notation, and boundary cases adopted by the library.
AI-generated: a genuinely novel statement formulated by AI, with no source for the claim itself.

These labels describe origin, not correctness: citations and verification chips remain separate evidence.

Every finite Borel measure on a compact Hausdorff space is regular

Statement

Assuming the Axiom of Countable Choice, the assertion “every finite Borel measure on a compact Hausdorff space is regular” is false.

Facts & Assumptions

Given: The Axiom of Countable Choice and $X = [0, ω_{1}]$ with the resulting Dieudonne probability measure $\overset{ˉ}{m}$ .

Refutation

technique · counterexample

1.1

The space $X$ is compact Hausdorff and $\overset{ˉ}{m} (X) = 1$ . For the open Borel set $Y = [0, ω_{1})$ , one has $\overset{ˉ}{m} (Y) = 1$ , while every compact $K \subseteq Y$ is bounded and has $\overset{ˉ}{m} (K) = 0$ .

given

2.1

Hence $\overset{ˉ}{m}$ is not even inner regular on the open set $Y$ , and therefore is not regular.

step 1.1∎

Depends on

Used by

The Dieudonne Borel measure on [0, omega₁] is not regular Counterexample

Dependency tree · two levels

33 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)