DefinitionDefinition: Literature-sourcedProof: Not applicableSession-authored (Fable 5 assisted)judge pass (gpt-5.6-terra)audited 2026-08-27

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The integral of a nonnegative simple function

Definition

Let $s = \sum_{j = 1}^{m} c_{j} χ_{E_{j}}$ be a simple representation of a nonnegative simple measurable function (Nonnegative simple measurable functions) on a measure space $(X, A, μ)$ , so the $E_{j} \in A$ are pairwise disjoint and $c_{j} \geq 0$ . Its simple integral is $\int s d μ : = \sum_{j = 1}^{m} c_{j} μ (E_{j}),$ where $μ$ is the given measure (Measures on sigma-algebras) and the convention $0 \cdot (+ \infty) = 0$ is fixed once and for all.

The next lemma proves that this value is independent of the chosen simple representation.

Depends on

Used by

The nonnegative Lebesgue integral Definition
The simple integral is independent of the chosen representation Lemma
The nonnegative integral agrees with the simple integral on simple functions Proposition
The simple integral is monotone, homogeneous, and additive Proposition

Dependency tree · two levels

6 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

John K. Hunter, Measure Theory Notes, Definition 4.1 (standard reference, not scraped)