Alphabeta Math
False statementConstruction: Literature-sourcedVerification: AI-generatedprecheck passjudge pass (gpt-5.6-terra)audited 2026-08-30
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FALSE: a Hahn decomposition is unique

Statement

False claim. Every signed measure has exactly one Hahn decomposition.

Facts & Assumptions

Given: The zero signed measure ν≡0 on the discrete measurable space (X,P(X)), where X={0,1}.

[L1]

Hahn decompositions are unique only up to null sets. (Hahn decomposition for signed measures, unique up to total-variation-null sets)

Refutation

technique · direct
1.1

Every measurable subset of X is both positive and negative for the zero [L1] measure, so both ∅⊔Xand{0}⊔{1} are Hahn decompositions.

2.1

These decompositions are different, so exact uniqueness fails. This is [L1, step 1.1] ∎ compatible with [L1] because the differing set is null.

Depends on

Used by

Nothing in the library uses this result yet.

Dependency tree · two levels

4 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