DefinitionDefinition: Literature-sourcedProof: Not applicablejudge 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.

Positive, negative, and null sets for a signed measure

Definition

Let $ν$ be a signed measure on $(X, A)$ .

A measurable set $P$ is positive for $ν$ when $ν (E) \geq 0$ for every measurable $E \subseteq P$ .
A measurable set $N$ is negative for $ν$ when $ν (E) \leq 0$ for every measurable $E \subseteq N$ .
A measurable set $Z$ is null for $ν$ when $ν (E) = 0$ for every measurable $E \subseteq Z$ .

The null-set clause is the strong one used throughout the page: it asks for vanishing on every measurable subset, not only on the ambient set itself.

Depends on

A signed measure is countably additive and takes at most one infinite value

Used by

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

Richard F. Bass, Real Analysis for Graduate Students, Definition 12.2 (standard reference, not scraped)
John K. Hunter, Measure Theory, Definition 6.16 (standard reference, not scraped)