DefinitionDefinition: Literature-sourcedProof: Not applicablejudge pass (gpt-5.6-terra)audited 2026-08-31

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.

A positive, signed, or complex measure concentrated on a measurable set

Definition

Let $ν$ be a positive measure, a signed measure, or a complex measure on $(X, A)$ , and let $S \in A$ . We say that $ν$ is concentrated on $S$ if $ν (E) = ν (E \cap S) (E \in A) .$ Equivalently, every measurable subset of $X ∖ S$ has $ν$ -value $0$ . For a positive measure this is the same as $ν (X ∖ S) = 0$ , but for a signed or complex measure the stronger subsetwise vanishing is the load-bearing form used in later proofs.

A positive, signed, or complex measure concentrated on a measurable set

Definition

Depends on

Dependency tree · 2 levels

Sources