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PropositionStatement: Literature-sourcedProof: AI-generatedprecheck passverified 2026-09-23 (gpt-6-sol)
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The nonnegative integral agrees with the simple integral on simple functions

Statement

If s is a nonnegative simple measurable function, then its nonnegative Lebesgue integral equals its simple integral: ∫s dμ=∫simples dμ.

Facts & Assumptions

Given: A nonnegative simple measurable function s.

[L1]

The nonnegative integral is the supremum of the simple integrals of all simple minorants 0≤u≤s (The nonnegative Lebesgue integral).

[L2]

The simple integral is monotone on nonnegative simple functions (The simple integral is monotone, homogeneous, and additive).

[L3]

The simple integral itself is well defined on every nonnegative simple function (The integral of a nonnegative simple function, The simple integral is independent of the chosen representation).

Proof

technique · direct
1.1givenL1L3

The function s is one of its own admissible simple minorants, so [L1] gives ∫s dμ≥∫simples dμ.

1.2L1L2L3

If u is any admissible simple minorant of s, then u≤s, so [L2] gives ∫simpleu dμ≤∫simples dμ. Taking the supremum over all such u in [L1] yields the reverse inequality.

2.1step 1.1step 1.2∎

The two inequalities from steps 1.1 and 1.2 give equality of the nonnegative and simple integrals on s.

Depends on

Dependency tree · 2 levels

Sources