PropositionStatement: Literature-sourcedProof: AI-adaptedprecheck passjudge pass (deepseek-v4-pro + claude-sonnet-5)audited 2026-08-17

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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.
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Every sigma-algebra is a lambda-system and a monotone class

Statement

Every sigma-algebra on $X$ is both a lambda-system and a monotone class on $X$ .

Facts & Assumptions

Given: A sigma-algebra $A$ on $X$ .

[L1]

A sigma-algebra contains the empty set and is closed under complements and countable unions (Sigma-algebras).

[L2]

A lambda-system contains $X$ , is closed under relative differences, and is closed under increasing countable unions (Lambda-systems, or Dynkin systems).

[L3]

A monotone class is closed under increasing countable unions and decreasing countable intersections (Monotone classes of sets).

Proof

technique · direct

1.1L1L2algebra

The family $A$ contains $X$ ; if $A \subseteq B$ lie in $A$ , then $B ∖ A = B \cap (X ∖ A)$ lies in $A$ ; and every increasing countable union lies in $A$ . Hence the axioms in [L2] hold.

2.1L1L3algebra∎

Increasing unions lie in $A$ by [L1]. If $(A_{n})$ decreases in $A$ , then $⋂_{n} A_{n} = X ∖ ⋃_{n} (X ∖ A_{n})$ lies in $A$ , so the axioms in [L3] hold.

Depends on

Used by

Nothing in the library uses this result yet.

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

A. Dembo, Probability Theory lecture notes, Definitions 1.1.36 and 1.1.43 (standard reference, not scraped)