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

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Nonnegative simple measurable functions

Definition

Let $(X, A, μ)$ be a measure space. A measurable function $s : X \to [0, + \infty) \subseteq \overline{R}$ (Extended-real-valued measurable functions) is a nonnegative simple measurable function when it has finite range.

Equivalently, there are pairwise disjoint measurable sets $E_{1}, \dots, E_{m} \in A$ and coefficients $c_{1}, \dots, c_{m} \in [0, + \infty)$ such that $s = \sum_{j = 1}^{m} c_{j} χ_{E_{j}} .$ Any such display is a simple representation of $s$ .

Depends on

Extended-real-valued measurable functions

Used by

The integral of a nonnegative simple function Definition
The nonnegative Lebesgue integral Definition
Every nonnegative measurable function is the increasing limit of simple measurable functions Theorem

Dependency tree · two levels

4 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)
Gerald B. Folland, Real Analysis, 2nd ed., §2.2 (standard reference, not scraped)