LemmaStatement: Literature-sourcedProof: AI-generatedprecheck passjudge pass (gpt-5.6-terra)audited 2026-09-05

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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.

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Measurable coordinatewise functions preserve independence

Statement

Let $(X_{i})_{i \in I}$ be an independent family of random elements $X_{i} : (Ω, F, P) \to (S_{i}, Σ_{i})$ . For each $i$ , let $g_{i} : (S_{i}, Σ_{i}) \to (T_{i}, T_{i})$ be measurable. Then the family $(g_{i} \circ X_{i})_{i \in I}$ is independent.

Facts & Assumptions

Given: Independent random elements $X_{i}$ and measurable maps $g_{i}$ as in the Statement.

[L1]

Measurable outer maps preserve measurability under composition. (Composition with a Borel measurable outer map preserves measurability)

[L2]

Independence of random elements is equivalent to the rectangle criterion. (Independent random elements are characterized by finite rectangle probabilities)

Proof

technique · direct

1.1

By [L1], each composite $g_{i} \circ X_{i}$ is again a random element.

givenL1

1.2

Fix a finite list of distinct indices $i_{0}, \dots, i_{n - 1}$ and measurable sets $C_{k} \in T_{i_{k}}$ . Then $g_{i_{k}}^{- 1} (C_{k}) \in Σ_{i_{k}}$ for every $k$ , so [L2] applied to the independent family $(X_{i})$ gives $P (g_{i_{0}} (X_{i_{0}}) \in C_{0}, \dots, g_{i_{n - 1}} (X_{i_{n - 1}}) \in C_{n - 1}) = \prod_{k < n} P (g_{i_{k}} (X_{i_{k}}) \in C_{k}) .$

givenL2

2.1

Step 1.2 is exactly the rectangle criterion for the family $(g_{i} \circ X_{i})_{i \in I}$ , so [L2] shows that this family is independent.

step 1.1step 1.2L2∎

Depends on

Used by

Dependency tree · two levels

9 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

Rick Durrett, Probability: Theory and Examples, 5th ed., Theorem 2.1.10 (standard reference, not scraped)
S. R. S. Varadhan, Probability Theory, Section 3.1 (standard reference, not scraped)