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.

Absolute continuity of a signed or complex measure with respect to a positive measure

Definition

Let $μ$ be a positive measure and let $ν$ be a signed measure or a complex measure on the same measurable space $(X, A)$ . We say that $ν$ is absolutely continuous with respect to $μ$ , and write $ν ≪ μ,$ if every measurable $μ$ -null set is also $ν$ -null: $μ (E) = 0 ⟹ ν (E) = 0 (E \in A) .$ Because every measurable subset of a $μ$ -null set is again $μ$ -null, this is equivalent to requiring $ν (F) = 0$ for every measurable $F \subseteq E$ whenever $μ (E) = 0$ .

Absolute continuity of a signed or complex measure with respect to a positive measure

Definition

Depends on

Dependency tree · 2 levels

Sources