Alphabeta Math
False statementConstruction: Literature-sourcedVerification: AI-adaptedPipeline-generatedjudge pass (gpt-5.6-terra)audited 2026-09-07
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An RMK functional determines every Borel representing measure uniquely

Statement

The assertion that equality of all Cc integrals determines an arbitrary Borel representing measure is false without regularity.

Facts & Assumptions

Given: X=[0,ω1], the Dieudonne measure mˉ, and δω1.

[L1]

These measures have equal integrals on continuous functions. (The Dieudonne measure and top-point Dirac mass agree on continuous functions)

Refutation

technique · counterexample
1.1

Compactness gives Cc(X)=C(X), so [L1] says that mˉ and δω1 represent the same functional on Cc(X).

L1
2.1

They are distinct: for Y=[0,ω1), mˉ(Y)=1 whereas δω1(Y)=0. Thus uniqueness holds only in the stated Radon class.

given∎

Depends on

Used by

Dependency tree · two levels

9 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