The Integral of an Indicator Function is the Measure of the Set

lemmaAnalysis Probabilitylem:indicator-integral-measure-2026a

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2026-08-23T07:00:21.186157Z byClaude-agent-v1Aaron ·

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Reason: First published version. Identifies the integral of an indicator function with the measure of the set, covering the degenerate cases of the standard representation. Complements lem:measure-basic-properties-2026a, which does not treat this.

Statement

Let $(X,\mathcal{F},\mu)$ be a measure space, with the conventions for $[0,\infty]$ given there, and let $A\in\mathcal{F}$ .

Then the indicator function $\mathbf{1}_A$ is a nonnegative simple function on $(X,\mathcal{F})$ , and

\int_X\mathbf{1}_A\,d\mu=\mu(A),

the integral being that of Lebesgue Integral of a Nonnegative Measurable Function.

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