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 families of event classes

Definition

Let $(Ω, F, P)$ be a probability space, let $I$ be an index set, and let $(C_{i})_{i \in I}$ be a family of classes of events $C_{i} \subseteq F$ .

The family $(C_{i})_{i \in I}$ is independent when for every natural number $n \geq 1$ , every choice of distinct indices $i_{0}, \dots, i_{n - 1} \in I$ , and every choice of events $A_{k} \in C_{i_{k}}$ , one has

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

Thus independence is a finite-subfamily condition. In particular, the empty family and every one-member family are independent.

Depends on

Probability measures and probability spaces

Used by

Independent Local Quantum Fault Model Definition
Independent sigma-algebras and independent events Definition
An empty risk set prevents root-n tightness Example
Weighted means from two independent subject groups Example
A random inner linear code has fewer than one bad word in expectation Lemma
Disjoint Fault-Location Events Are Independent Lemma
Independent families pass to subfamilies Lemma
The finite product law has iid coordinate maps Lemma
IID terminal influence limit from a separately established expansion Theorem
Independent-group terminal influence limit Theorem
PCP soundness amplification by independent repetition Theorem
Planar Surface-Code Memory Bound Under IID Z Noise Theorem
Terminal normal limit for weighted logrank scores Theorem

Dependency tree · two levels

3 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)