Alphabeta Math
False statementConstruction: Literature-sourcedVerification: AI-generatedprecheck passjudge pass (gpt-5.6-terra)audited 2026-08-30
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  • 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.
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FALSE: total variation always equals the absolute value of the set value

Statement

False claim. For every signed measure or complex measure and every measurable set E, one has ∣ν∣(E)=∣ν(E)∣.

Facts & Assumptions

Given: The signed measure ν=δ1−δ−1 on the discrete measurable space (E,P(E)), where E={−1,1}.

[L1]

The Dirac set function at a point is indicator-valued. (The Dirac set function at a point)

[L2]

Total variation is the supremum of measurable partition sums. (The total variation |nu|(E) from countable measurable partitions)

Refutation

technique · direct
1.1

By [L1], one has ν(E)=0, so ∣ν(E)∣=0.

L1
2.1

The partition E={1}⊔{−1} gives partition sum [L1, L2, step 1.1] ∎ ∣ν({1})∣+∣ν({−1})∣=2, so [L2] yields ∣ν∣(E)≥2. Hence ∣ν∣(E)≠∣ν(E)∣.

Depends on

Used by

Nothing in the library uses this result yet.

Dependency tree · two levels

6 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