TheoremStatement: AI-adaptedProof: AI-adaptedPipeline-generatedjudge pass (gpt-5.6-terra)audited 2026-09-10

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Integral invariance under measure-preserving maps

Statement

If $T$ preserves $μ$ and $f : X \to [0, \infty]$ is measurable, then $\int f \circ T d μ = \int f d μ$ , allowing infinity. If $f$ is integrable real or complex valued, $f \circ T$ is integrable and the same equality holds. Conversely, for a measurable self-map, equality for every measurable indicator implies measure preservation.

Facts & Assumptions

[F1]

Nonnegative measurable functions admit increasing simple approximations Every nonnegative measurable function is the increasing limit of simple measurable functions.

[F2]

Increasing nonnegative measurable functions have increasing integrals with the expected limit Monotone convergence for the integral.

[F3]

The integral is complex-linear on integrable functions The Lebesgue integral is linear on $L^{1} (μ)$ .

Proof

Given: The objects and hypotheses in the statement.

1.1

For $E \in A$ , $1_{E} \circ T = 1_{T^{- 1} E}$ , hence its integral is $μ (T^{- 1} E) = μ (E)$ . A nonnegative simple function written over disjoint fibers has integral equal to the sum of coefficient times fiber measure, so the identity holds for every such function, also when the sum is infinite in value.

given

2.1

For nonnegative measurable $f$ , take $s_{n} ↑ f$ as supplied by simple approximation. Then $s_{n} \circ T ↑ f \circ T$ measurably. Integral monotone convergence on both sides gives $\int f \circ T = \lim_{n} \int s_{n} \circ T = \lim_{n} \int s_{n} = \int f$ .

F1F2step 1.1

3.1

For integrable $f$ , applying step 2.1 to $∣ f ∣$ proves $\int ∣ f \circ T ∣ = \int ∣ f ∣ < \infty$ . Apply that step to the positive and negative parts of each real component. Subtract their finite integrals and combine the real and imaginary parts by linearity to obtain the asserted equality.

F3step 2.1

4.1

Conversely the indicator identity is exactly $μ (T^{- 1} E) = μ (E)$ for each measurable $E$ . Thus it is measure preservation.

step 1.1given∎

Depends on

Used by

Dependency tree · two levels

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Sources

E–W Lemma 2.6, pp.15–16 (standard reference, not scraped)