DefinitionDefinition: Literature-sourcedProof: Not applicablejudge 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 sigma-algebras and independent events

Definition

Let $(Ω, F, P)$ be a probability space.

A family $(F_{i})_{i \in I}$ of sub-sigma-algebras of $F$ is independent when it is independent as a family of event classes in the sense of Independent families of event classes.

A family $(A_{i})_{i \in I}$ of events is independent when the singleton event classes $({A_{i}})_{i \in I}$ are independent. Equivalently, for every finite choice of distinct indices $i_{0}, \dots, i_{n - 1}$ ,

$P (⋂_{k < n} A_{i_{k}}) = \prod_{k < n} P (A_{i_{k}}) .$

For two events this reduces to the familiar identity $P (A \cap B) = P (A) P (B)$ .

Depends on

Used by

Independent random elements Definition
Pairwise independence Definition
Independent events remain independent under complements Lemma
Disjoint groups of an independent sigma-algebra family remain independent Theorem
Independent pi-systems generate independent sigma-algebras Theorem

Dependency tree · two levels

4 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)
S. R. S. Varadhan, Probability Theory, Section 3.1 (standard reference, not scraped)