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

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The canonical coordinate process realizes consistent finite-dimensional laws

Statement

Under the extension measure of Assuming the Axiom of Choice, Kolmogorov extension for arbitrary families of standard Borel coordinate spaces, $X_{i} (x) = x_{i}$ is a process whose finite-dimensional distributions are the prescribed $(μ_{F})$ .

Facts & Assumptions

Given: The extension measure $μ$ on the cylinder space and its coordinate maps.

[F1]

The extension has marginal $μ_{F}$ under every finite-coordinate projection. (Assuming the Axiom of Choice, Kolmogorov extension for arbitrary families of standard Borel coordinate spaces)

Proof

1.1

Each coordinate map is measurable because inverse images of its measurable sets are one-coordinate cylinders. Thus $(X_{i})$ is a process.

2.1

For finite $F$ , $(X_{i})_{i \in F} = π_{F}$ , so its pushforward law is $μ_{F}$ by [F1]. This is precisely the finite-dimensional-law convention.

F1∎

Depends on

Used by

An uncountable Bernoulli coordinate process Example

Dependency tree · two levels

18 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

Biskup, MATH 275D notes, Theorem 2.4 (standard reference, not scraped)