LemmaStatement: Literature-sourcedProof: AI-generatedPipeline-generatedjudge pass (gpt-5.6-terra)audited 2026-09-06

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Finite-coordinate cylinders form a $π$ -system

Statement

The finite-coordinate cylinders in $E = \prod_{i \in I} E_{i}$ form a pi-system.

Facts & Assumptions

Given: Two finite-coordinate cylinders $π_{F}^{- 1} (A)$ and $π_{G}^{- 1} (B)$ .

[F1]

Their supports $F, G$ are finite, and their bases are measurable in the respective finite product sigma-algebras.

Proof

1.1F1

Put $H = F \cup G$ and let $p_{F}, p_{G}$ be the coordinate projections from $\prod_{H} E_{i}$ . Then $C : = p_{F}^{- 1} (A) \cap p_{G}^{- 1} (B)$ is measurable in $⨂_{H} E_{i}$ .

2.1step 1.1∎

Directly from restriction of coordinates, $π_{F}^{- 1} (A) \cap π_{G}^{- 1} (B) = π_{H}^{- 1} (C)$ , a finite-coordinate cylinder. The family is nonempty because it contains $E$ , so it is a pi-system.

Depends on

Dependency tree · 2 levels

Sources

Biskup, MATH 275D notes, Lemma 2.6

Finite-coordinate cylinders form a ππ-system