PropositionStatement: AI-adaptedProof: AI-adaptedPipeline-generatedjudge pass (gpt-5.6-terra)audited 2026-09-13

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First-return time and induced map are measurable

Statement

With $E, r_{E}, E_{\infty}, T_{E}$ as in the first-return definition, $r_{E} : E \to {1, 2, \dots} \cup {\infty}$ is measurable for the full power-set sigma-algebra on this countable target, and $T_{E}$ is measurable for $A ∣_{E_{\infty}}$ .

Facts & Assumptions

[F1]

The core is measurable and T_E is an everywhere defined self-map of it. First-return times and induced transformations.

[F2]

Every finite iterate is measurable on the original sigma-algebra. Compositions, iterates and completions preserve invariance.

[F3]

Trace sets are intersections with the domain. The trace of a sigma-algebra on a subset.

Proof

Given: With $E, r_{E}, E_{\infty}, T_{E}$ as in the first-return definition, $r_{E} : E \to {1, 2, \dots} \cup {\infty}$ is measurable for the full power-set sigma-algebra on this countable target, and $T_{E}$ is measurable for $A ∣_{E_{\infty}}$ .

1.1

For $n \geq 1$ , the fiber $H_{n} = {x \in E : r_{E} (x) = n} = E \cap T^{- n} E \cap ⋂_{1 \leq j < n} T^{- j} (X ∖ E)$ is measurable. At $n = 1$ the empty intersection is $X$ . The fiber at infinity is $E ∖ ⋃_{n \geq 1} H_{n}$ . Every inverse image of a subset of the countable target is a countable union of these fibers, proving measurability of $r_{E}$ .

F1F2

2.1

For $B \in A ∣_{E_{\infty}}$ the measurability of $E_{\infty}$ implies $B \in A$ . The identity $T_{E}^{- 1} B = ⋃_{n \geq 1} (E_{\infty} \cap H_{n} \cap T^{- n} B)$ expresses its inverse image as a measurable subset of $E_{\infty}$ , hence a trace set. Empty B gives an empty union of pieces; full B gives all of the core.

step 1.1F1F2F3∎

Depends on

Used by

Induced transformations preserve restricted finite measure Theorem
Kac integral formula for excursions Theorem
Kac return-time formula without invertibility Theorem

Dependency tree · two levels

13 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

Sarig §1.6.4 p.28 (standard reference, not scraped)