DefinitionDefinition: Literature-sourcedProof: Not applicablejudge pass (gpt-5.6-terra)audited 2026-09-05

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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
L2-m-approximable causal Bernoulli shifts Definition
Pairwise independence Definition
Finite-dependent functional moving average Example
Competing exponential clocks give the reaction event law Lemma
Conditioning a known state and independent noise Lemma
Elementary Ito integrals do not depend on step representation Lemma
Finite-dependent vector central limit theorem Lemma
Independent events remain independent under complements Lemma
Independent noise extends and pushes forward a conditional law Lemma
Regular conditional kernels push forward under measurable random maps Lemma
The finite product law has iid coordinate maps Lemma
The monotone IV first stage is the complier proportion Lemma
An independent pilot coefficient preserves conditional unbiasedness Theorem
Conformal rank superuniformity under exchangeability Theorem
Disjoint groups of an independent sigma-algebra family remain independent Theorem
Empirical-law bootstrap draws are conditionally IID Theorem
Fixed-order autoregressive OLS martingale limit Theorem
Full conformal marginal coverage Theorem
Future-path Markov property Theorem
Gaussian simple kriging is a conditional mean Theorem
Independent pi-systems generate independent sigma-algebras Theorem
Ito isometry for elementary integrands Theorem
Markov property of Brownian motion Theorem
Split conformal marginal coverage 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)