Alphabeta Math
TheoremStatement: 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-dimensional distributions determine a process law on the cylinder sigma-algebra

Statement

If two probability measures on CI have equal finite-dimensional marginals, then they are equal.

Facts & Assumptions

Given: Probability measures μ,ν on CI with equal finite-dimensional marginals.

[F1]

Finite-coordinate cylinders form a pi-system generating CI. (Finite-coordinate cylinders form a π-system)

[F2]

A lambda-system containing a pi-system contains the sigma-algebra it generates. (Dynkin's pi-lambda theorem)

Proof

1.1

On D={C∈CI:μ(C)=ν(C)}, complements and disjoint countable unions preserve equality because both measures are probabilities. Thus D is a lambda-system.

F2
2.1

Equal finite-dimensional marginals put every cylinder in D. By [F1] and [F2], CI⊆D, hence μ=ν.

F1F2∎

Depends on

Used by

Nothing in the library uses this result yet.

Dependency tree · two levels

14 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