CounterexampleConstruction: 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.

The Dieudonne Borel measure on [0, omega_1] is not regular

Statement refuted

Assuming the Axiom of Countable Choice, a finite Borel measure on a compact Hausdorff space need not be regular.

Facts & Assumptions

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

Counterexample

technique · direct

1.1

The space $X$ is compact Hausdorff and $\overset{ˉ}{m} (X) = 1$ . The open Borel subset $Y = [0, ω_{1})$ contains a club, so $\overset{ˉ}{m} (Y) = 1$ .

given

2.1

Every compact $K \subseteq Y$ is bounded below $ω_{1}$ , hence contains no club and has $\overset{ˉ}{m} (K) = 0$ . Therefore $Y$ cannot be approximated from within by compact sets, and $\overset{ˉ}{m}$ is not regular.

given∎

Depends on

Used by

Distinct Borel measures can represent the same C_c functional 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)