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

How statement and proof provenance work

The first chip identifies the source of the statement or construction; the second identifies the source of its local proof or verification.

Literature-sourced: the exact statement appears in a cited source; only wording and notation differ.
AI-adapted: a semantically identical restatement of literature-sourced material, modulo indexing, notation, and boundary cases adopted by the library.
AI-generated: a genuinely novel statement formulated by AI, with no source for the claim itself.

These labels describe origin, not correctness: citations and verification chips remain separate evidence.

Finite-coordinate cylinder sets form an algebra

Statement

The family $A_{I}$ of all finite-coordinate cylinders in $E$ is an algebra of subsets of $E$ .

Facts & Assumptions

Given: Cylinders $π_{F}^{- 1} (A)$ and $π_{G}^{- 1} (B)$ .

[F1]

A finite product sigma-algebra is closed under complements, inverse images under coordinate projections, and finite unions.

Proof

1.1

The empty-support cylinders give $\emptyset, E \in A_{I}$ . Also $E ∖ π_{F}^{- 1} (A) = π_{F}^{- 1} ((\prod_{F} E_{i}) ∖ A) \in A_{I}$ .

2.1

For $H = F \cup G$ , pull both bases to $\prod_{H} E_{i}$ ; their union is measurable and its inverse image is $π_{F}^{- 1} (A) \cup π_{G}^{- 1} (B)$ . Thus $A_{I}$ has the three algebra operations.

F1∎

Depends on

Used by

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

Dependency tree · two levels

5 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

Kajino, Probability Theory, proof of Theorem 3.65 (standard reference, not scraped)