LemmaStatement: AI-adaptedProof: AI-generatedPipeline-generatedverified 2026-10-15 (gpt-6-luna, gpt-6.1-sol, and delegated Codex review/repair agents; see completed run evidence)

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The thermodynamics measure integral agrees with the shared nonnegative integral

Statement

Let $(X, Σ, μ)$ be a measure space and let $f : X \to [0, + \infty]$ be $Σ$ -measurable. The nonnegative integral defined by Measure space and integral equals the shared nonnegative Lebesgue integral defined by The nonnegative Lebesgue integral, with the same value in $[0, + \infty]$ .

Facts & Assumptions

Given: A measure space $(X, Σ, μ)$ and a nonnegative measurable function $f$ .

[F1]

Measure space and integral defines a nonnegative simple integral by finite sums $\sum_{j} c_{j} μ (A_{j})$ , with $0 \cdot (+ \infty) = 0$ , and defines $\int f d μ$ as the supremum over measurable simple minorants $0 \leq s \leq f$ . Its current batch-11 carrier is a draft definition; no proof is required for this definition-like supplier.

[F2]

The nonnegative Lebesgue integral defines the shared nonnegative integral as the supremum over the same class of nonnegative simple minorants.

[F3]

Nonnegative simple measurable functions characterizes nonnegative simple functions as finite-range measurable maps, and A simple function and its canonical representation gives each such function its finite partition into level sets.

[F4]

Measures on sigma-algebras supplies the countably additive measure $μ$ in the measure space. The integral of a nonnegative simple function uses the finite sum $\sum_{j} c_{j} μ (A_{j})$ with the convention $0 \cdot (+ \infty) = 0$ ; The simple integral is independent of the chosen representation proves that this value does not depend on the chosen simple representation.

Proof

1.1F1F2F3construct

A nonnegative simple function in the thermodynamics definition is a finite sum of indicators of disjoint measurable sets, hence is measurable and has finite range; conversely, the level sets in [F3] represent every nonnegative finite-range measurable function in that form. Thus the two definitions use the same class of simple minorants.

2.1F1F3F4step 1.1

For any simple minorant $s$ , use its canonical level-set representation from [F3]. Both [F1] and [F4] assign the same finite sum $\sum_{j} c_{j} μ (A_{j})$ to $s$ , including zero coefficients on sets of infinite measure; [F4] ensures the shared value is representation-independent. Hence the two definitions assign the same value to every minorant.

3.1F1F2step 1.1step 2.1∎

The two nonnegative integrals are suprema over the same minorants with the same value for each minorant by steps 1.1 and 2.1. Their suprema therefore agree, including when the common value is $+ \infty$ .

The thermodynamics measure integral agrees with the shared nonnegative integral

Statement

Facts & Assumptions

Proof

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