DefinitionDefinition: Literature-sourcedProof: Not applicablePipeline-generatedjudge pass (gpt-5.6-terra)audited 2026-09-06

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.

Indefinite Lebesgue integral on a compact interval

Let $f \in L^{1} ([a, b])$ in the sense of Integrable real and complex functions, and their integrals. Its indefinite Lebesgue integral based at $a$ is $I_{f} (x) : = \int_{[a, x]} f d λ = \int_{a}^{x} f (t) d t (a \leq x \leq b),$ where, for real- or complex-valued $f$ , the first integral means $\int_{[a, b]} f 1_{[a, x]} d λ$ in the sense of Integrable real and complex functions, and their integrals. For nonnegative $f$ this agrees with the set-integral notation of Integral over a measurable subset. Thus $I_{f} (a) = 0$ , including when $a = b$ .

Depends on