LemmaStatement: 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.
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A lambda-system closed under finite intersections is a sigma-algebra

Statement

If a lambda-system $D$ on $X$ is closed under binary intersections, then $D$ is a sigma-algebra on $X$ .

Facts & Assumptions

Given: A lambda-system $D$ on $X$ that is closed under binary intersections.

[L1]

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

[L2]

A sigma-algebra is an algebra closed under countable unions (Sigma-algebras).

Proof

technique · direct

1.1L1

Since $X \in D$ , [L1] gives $X ∖ A \in D$ for every $A \in D$ .

2.1step 1.1givenalgebra

For $A, B \in D$ , step 1.1 and intersection closure give $A \cup B = X ∖ ((X ∖ A) \cap (X ∖ B)) \in D$ . Thus every finite union of members belongs to $D$ .

3.1step 1.1step 2.1L1L2∎

For a sequence $(A_{n})$ in $D$ , the partial unions $B_{n} : = ⋃_{k \leq n} A_{k}$ lie in $D$ by step 2.1 and increase, so [L1] gives $⋃_{n} A_{n} = ⋃_{n} B_{n} \in D$ . Together with steps 1.1 and 2.1, this is the sigma-algebra criterion [L2].

Depends on

Used by

Dynkin's pi-lambda theorem 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

A. Dembo, Probability Theory lecture notes, Proposition 1.1.37 (standard reference, not scraped)