DefinitionDefinition: Literature-sourcedProof: Not applicablejudge pass (gpt-5.6-terra)audited 2026-08-27

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.

Integral over a measurable subset

Definition

Let $(X, A, μ)$ be a measure space, let $f : X \to [0, + \infty]$ be measurable, and let $E \in A$ . Since Closure properties of measurable functions used by the integral implies that $f χ_{E}$ is measurable, define the integral of $f$ over $E$ by $\int_{E} f d μ : = \int f χ_{E} d μ,$ where the integral on the right is the nonnegative Lebesgue integral of The nonnegative Lebesgue integral.

Depends on

Dependency tree · 2 levels

Sources

John K. Hunter, Measure Theory Notes, Definition 4.4