Alphabeta Math
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.

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

Countably many independent copies of a prescribed law exist

Statement

Assume countable choice and dependent choice. Every probability measure ν on (S,Σ) is the common law of a countable independent family of S-valued random elements.

Facts & Assumptions

Given: Countable choice, dependent choice, and a measurable probability space (S,Σ,ν).

[F1]

Under countable choice and dependent choice, the canonical countable product probability measure exists. (Assuming countable and dependent choice, countable products of arbitrary probability spaces)

[F2]

Its canonical coordinates have their prescribed laws and are independent. (Coordinate random elements of a countable product are independent)

Proof

1.1

Apply [F1] with En=S and μn=ν for every n, and write Xn for its coordinate maps.

F1
2.1

By [F2], every Xn has law ν and every finite subfamily is independent; this is the required independent-copy sequence.

F2∎

Depends on

Used by

Nothing in the library uses this result yet.

Dependency tree · two levels

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Sources