CounterexampleConstruction: AI-generatedVerification: AI-generatedprecheck passjudge pass (gpt-5.6-terra)audited 2026-08-30

How statement and proof provenance work

The first chip identifies the source of the statement or construction; the second identifies the source of its local proof or verification.

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.
AI-generated: a genuinely novel statement formulated by AI, with no source for the claim itself.

These labels describe origin, not correctness: citations and verification chips remain separate evidence.

Total variation can exceed the absolute value of the set value

Statement refuted

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

Facts & Assumptions

Given: The signed measure $ν = δ_{1} - δ_{- 1}$ on the discrete measurable space $(E, P (E))$ , where $E = {- 1, 1}$ .

[L1]

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

[L2]

Total variation is the supremum of countable partition sums of $∣ ν (E_{n}) ∣$ . (The total variation |nu|(E) from countable measurable partitions)

Counterexample

technique · direct

1.1

By [L1], one has $ν (E) = δ_{1} (E) - δ_{- 1} (E) = 1 - 1 = 0$ , so [L1] $∣ ν (E) ∣ = 0$ .

2.1

The two singletons ${1}$ and ${- 1}$ form a measurable partition of [L1, L2, step 1.1] ∎ $E$ , and [L1] gives $∣ ν ({1}) ∣ + ∣ ν ({- 1}) ∣ = 1 + 1 = 2.$ Therefore [L2] yields $∣ ν ∣ (E) \geq 2 > 0 = ∣ ν (E) ∣$ , refuting the claim.

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

Richard F. Bass, Real Analysis for Graduate Students, Chapter 12 (standard reference, not scraped)