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

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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.

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Arbitrary products of standard Borel probability spaces

Statement

Assume AC. For standard-Borel probability spaces $(E_{i}, E_{i}, μ_{i})_{i \in I}$ there is a unique probability measure on $C_{I}$ whose finite-coordinate marginals are the finite product measures $⨂_{i \in F} μ_{i}$ .

Facts & Assumptions

Given: AC and a family of standard-Borel probability spaces.

[F1]

Finite product measures have the rectangle formula and are probability measures. (For sigma-finite factors, the product measure exists, has the rectangle formula, is sigma-finite, and is unique)

[F2]

Kolmogorov extension applies to every consistent finite-dimensional family on standard-Borel coordinates. (Assuming the Axiom of Choice, Kolmogorov extension for arbitrary families of standard Borel coordinate spaces)

Proof

1.1

For finite $F$ , define $ν_{F} = ⨂_{i \in F} μ_{i}$ . The rectangle formula shows that projecting $ν_{G}$ to $F \subseteq G$ gives $ν_{F}$ , so the family is consistent.

2.1

Apply [F2] to $(ν_{F})$ . Its conclusion is exactly the stated measure and its cylinder-sigma uniqueness; it does not assert a measure on $P (\prod_{i} E_{i})$ .

F2∎

Depends on

Used by

Nothing in the library uses this result yet.

Dependency tree · two levels

23 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

Shalizi, Building Processes, Theorem 29 (standard reference, not scraped)