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

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Measure preservation can be checked on a generating pi-system

Statement

Let $T : X \to X$ be measurable on $(X, A, μ)$ . Let $P$ be a pi-system generating $A$ , with an increasing sequence $P_{n} \in P$ covering $X$ and satisfying $μ (P_{n}) < \infty$ . If $μ (T^{- 1} P) = μ (P)$ for every $P \in P$ , then $T$ preserves $μ$ . For finite $μ$ , a generating pi-system can be enlarged by $X$ to meet the exhaustion condition.

Facts & Assumptions

[F1]

Two measures agreeing on a generating pi-system and an increasing finite-mass exhaustion agree everywhere Measures agreeing on a generating pi-system are equal under an increasing finite-measure exhaustion from that pi-system.

Proof

Given: The objects and hypotheses in the statement.

1.1

Set $ν (E) = μ (T^{- 1} E)$ . Inverse images take the empty set to the empty set and disjoint countable unions to disjoint countable unions. Thus $ν (\emptyset) = 0$ and $ν (⋃_{j} E_{j}) = \sum_{j} ν (E_{j})$ for disjoint measurable $E_{j}$ , so $ν$ is a measure.

given

2.1

The measures $ν, μ$ agree on $P$ , and $ν (P_{n}) = μ (P_{n}) < \infty$ on the stated increasing exhaustion. Uniqueness therefore gives $ν (E) = μ (E)$ for all $E \in A$ , which is measure preservation.

F1step 1.1

3.1

If $μ (X) < \infty$ , adjoining $X$ keeps the family a generating pi-system; its preservation identity holds since $T^{- 1} X = X$ . The constant exhaustion $P_{n} = X$ then satisfies every required hypothesis.

step 2.1given∎

Depends on

Used by

Dependency tree · two levels

9 results within two dependency steps of this one, each drawn at its shortest distance from it. An arrow runs from a result to what uses it, so the chart reads left to right and ends at this result, which carries a heavier outline. Every node is a link to that result. Click elsewhere on the chart to enlarge it.

Sources

E–W §2.1 p.13; local sigma-finite uniqueness theorem (standard reference, not scraped)