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

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Distinct Borel measures can represent the same C_c functional

Statement refuted

Without a Radon or regularity hypothesis, a functional on $C_{c} (X)$ need not determine its Borel representing measure uniquely.

Facts & Assumptions

Given: On $X = [0, ω_{1}]$ , let $\overset{ˉ}{m}$ be the Dieudonne measure and let $δ = δ_{ω_{1}}$ .

Counterexample

technique · direct

1.1

Since $X$ is compact, $C_{c} (X) = C (X)$ . Eventual constancy gives $\int f d \overset{ˉ}{m} = f (ω_{1}) = \int f d δ$ for every $f \in C_{c} (X)$ .

given

2.1

Yet for $Y = [0, ω_{1})$ , $\overset{ˉ}{m} (Y) = 1$ and $δ (Y) = 0$ . Thus the two Borel measures are distinct representations of the same functional; the missing condition is regularity.

given∎

Depends on

Used by

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Dependency tree · two levels

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Sources

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