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

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Consistent finite-dimensional laws define a well-defined finitely additive cylinder law

Statement

Suppose the product $E = \prod_{i \in I} E_{i}$ is nonempty. For a consistent family $(μ_{F})$ , the formula $μ_{0} (π_{F}^{- 1} (A)) : = μ_{F} (A)$ is well-defined on $A_{I}$ and is finitely additive.

Facts & Assumptions

Given: A nonempty product $E$ , a consistent family $(μ_{F})$ , and a finite disjoint cylinder decomposition.

[F1]

If $F \subseteq H$ , consistency says $(p_{H, F})_{#} μ_{H} = μ_{F}$ .

[F2]

Finite-coordinate cylinders form an algebra. (Finite-coordinate cylinder sets form an algebra)

Proof

1.1

Fix $x^{*} \in E$ . If $π_{F}^{- 1} (A) = π_{G}^{- 1} (B)$ , pull both sets to $H = F \cup G$ . Their lifted bases are equal: otherwise a point of their symmetric difference, combined with the coordinates of $x^{*}$ outside $H$ , would distinguish the cylinders. Hence consistency gives $μ_{F} (A) = μ_{H} (p_{H, F}^{- 1} A) = μ_{H} (p_{H, G}^{- 1} B) = μ_{G} (B)$ .

2.1

For disjoint cylinders $C_{r} = π_{F_{r}}^{- 1} (A_{r})$ with $r < m$ , pull all bases to $H = ⋃_{r < m} F_{r}$ . They are disjoint measurable sets, so finite additivity of $μ_{H}$ yields $μ_{0} (⋃_{r} C_{r}) = \sum_{r} μ_{0} (C_{r})$ . The union is in the cylinder algebra by [F2].

F1F2∎

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Sources

Biskup, MATH 275D notes, Lemma 2.6 (standard reference, not scraped)