DefinitionDefinition: Literature-sourcedProof: Not applicableSession-authored (Fable 5 assisted)judge pass (deepseek-v4-pro + claude-sonnet-5)audited 2026-08-17

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.

Monotone classes of sets

Definition

Let $X$ be a set. A monotone class on $X$ is a family $M \subseteq P (X)$ satisfying both closure conditions:

if $A_{0} \subseteq A_{1} \subseteq \dots$ and every $A_{n} \in M$ , then $⋃_{n \in N} A_{n} \in M$ ;
if $A_{0} \supseteq A_{1} \supseteq \dots$ and every $A_{n} \in M$ , then $⋂_{n \in N} A_{n} \in M$ .

The sequences are indexed by $N$ beginning at $0$ .

Used by

The monotone class generated by a family of sets Definition
FALSE: every monotone class is an algebra False statement
FALSE: the union of an increasing sequence of monotone classes is a monotone class False statement
Every sigma-algebra is a lambda-system and a monotone class Proposition

Dependency tree · next 3 levels

Nothing. This result depends on no other item in the library.

Sources

R. F. Bass, Real Analysis for Graduate Students, version 5.0, Definition 2.9 (standard reference, not scraped)
A. Dembo, Probability Theory lecture notes, Definition 1.1.43 (standard reference, not scraped)