DefinitionDefinition: Literature-sourcedProof: Not applicableaudited 2026-08-27

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The nonnegative Lebesgue integral

Definition

Let $f : X \to [0, + \infty]$ be measurable (Extended-real-valued measurable functions). Its nonnegative Lebesgue integral is $\int f d μ : = \sup {\int s d μ : s is nonnegative simple and 0 \leq s \leq f},$ where the simple integral on the right is the one from The integral of a nonnegative simple function.

The set of admissible simple minorants is nonempty because it contains the zero function, which is simple (Nonnegative simple measurable functions).

Definition references

Extended-real-valued measurable functions

The integral of a nonnegative simple function

Nonnegative simple measurable functions

The nonnegative Lebesgue integral

Definition

Definition references

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