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

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Literature-sourced: the exact statement appears in a cited source; only wording and notation differ.
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The Dieudonne measure and top-point Dirac mass agree on continuous functions

Statement

Assume the Axiom of Countable Choice. For the Dieudonne measure $\overset{ˉ}{m}$ on $[0, ω_{1}]$ and the Dirac measure $δ_{ω_{1}}$ , $\int f d \overset{ˉ}{m} = f (ω_{1}) = \int f d δ_{ω_{1}}$ for every continuous real or complex $f$ .

Facts & Assumptions

Given: The Axiom of Countable Choice and the resulting extended club-set probability measure $\overset{ˉ}{m}$ .

[L1]

Continuous functions on $[0, ω_{1}]$ are eventually constant. (Continuous functions on [0, omega_1] are eventually constant)

Proof

technique · direct

1.1

By [L1], $f = c = f (ω_{1})$ on a tail $[α, ω_{1}]$ . Its intersection with $Y = [0, ω_{1})$ contains a club, so its complement has $\overset{ˉ}{m}$ -measure zero. Thus $f = c$ almost everywhere for $\overset{ˉ}{m}$ .

2.1

Since both measures are probabilities, $\int f d \overset{ˉ}{m} = c$ , while the defining property of a Dirac measure gives $\int f d δ_{ω_{1}} = f (ω_{1}) = c$ .

step 1.1∎

Depends on

Used by

Distinct Borel measures can represent the same C_c functional Counterexample
An RMK functional determines every Borel representing measure uniquely False statement

Dependency tree · two levels

16 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)