CorollaryStatement: 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.

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

1.1F1

Apply [F1] with $E_{n} = S$ and $μ_{n} = ν$ for every $n$ , and write $X_{n}$ for its coordinate maps.

2.1F2∎

By [F2], every $X_{n}$ has law $ν$ and every finite subfamily is independent; this is the required independent-copy sequence.

Depends on

Dependency tree · 2 levels

Sources

Durrett, Probability: Theory and Examples, Section 2.1.4