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

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Indefinite Lebesgue integral on a compact interval

Definition

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

Used by

Dependency tree · two levels

5 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

Donald L. Cohn, Measure Theory, 2nd ed., §6.3 (standard reference, not scraped)