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

Used by

Birkhoff strong law for iid coordinate shifts Corollary
Countably many independent copies of a prescribed law exist Corollary
Summable untruncated variances are not necessary Counterexample
A canonical random walk from product increments Example
A Lindeberg array with no identically distributed row Example
A nonidentical Bernoulli weak law Example
Almost sure frequency of heads Example
An i.i.d. sequence with a prescribed law Example
Independent but non-identically distributed coordinates Example
The infinite fair-coin-toss space Example
The three series impose separate conditions Example
Independent Brownian motion straightens a Poisson clock Lemma
Disjoint streams of independent fair bits produce independent uniform variables Theorem
Rejection sampling is uniform, almost surely terminating, and has expected linear bit cost Theorem

Dependency tree · two levels

15 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

Kajino, Probability Theory, Proposition 3.67 (standard reference, not scraped)