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

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.

Independent families pass to subfamilies

Statement

Let $(C_{i})_{i \in I}$ be an independent family of event classes on a probability space, and let $J \subseteq I$ . Then the subfamily $(C_{j})_{j \in J}$ is independent.

Facts & Assumptions

Given: An independent family $(C_{i})_{i \in I}$ and a subset $J \subseteq I$ .

[L1]

Independence means that every finite choice of distinct indices and one event from each chosen class satisfies the product formula (Independent families of event classes).

Proof

technique · direct

1.1

Fix a natural number $n \geq 1$ , distinct indices $j_{0}, \dots, j_{n - 1} \in J$ , and events $A_{k} \in C_{j_{k}}$ . Since $J \subseteq I$ , this is also a valid finite choice inside the original family.

givenL1

2.1

Applying [L1] to that same finite choice gives $P (⋂_{k < n} A_{k}) = \prod_{k < n} P (A_{k}) .$ Hence the restricted family is independent.

step 1.1L1∎

Depends on

Independent families of event classes

Used by

Nothing in the library uses this result yet.

Dependency tree · two levels

2 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., Section 2.1 (standard reference, not scraped)