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

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Coordinate random elements of a countable product are independent

Statement

Under the measure of Assuming countable and dependent choice, countable products of arbitrary probability spaces, the coordinate maps $X_{n} (x) = x_{n}$ have laws $μ_{n}$ and are independent.

Facts & Assumptions

Given: The canonical countable product probability space and finitely many distinct indices $n_{1}, \dots, n_{k}$ .

[F1]

The product measure has every prescribed finite product marginal. (Assuming countable and dependent choice, countable products of arbitrary probability spaces)

[F2]

Independence of random elements means the finite intersection formula for their measurable inverse images. (Independent random elements)

Proof

1.1F1

For $A_{j} \in E_{n_{j}}$ and the distinct indices fixed above, [F1] gives $P (⋂_{j} {X_{n_{j}} \in A_{j}}) = \prod_{j} μ_{n_{j}} (A_{j})$ ; taking one coordinate gives the asserted law.

2.1F2∎

The right side is $\prod_{j} P (X_{n_{j}} \in A_{j})$ , which is exactly [F2]. Since the finite family was arbitrary, all coordinates are independent.

Depends on

Dependency tree · 2 levels

Sources

Kajino, Probability Theory, Proposition 3.67