TheoremStatement: Literature-sourcedProof: AI-adaptedSession-authored (Fable 5 assisted)precheck passjudge 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.

Dynkin's pi-lambda theorem

Statement

Let $P$ be a pi-system on $X$ . Then $λ_{X} (P) = σ_{X} (P)$ . Consequently, if $D$ is any lambda-system on $X$ with $P \subseteq D$ , then $σ_{X} (P) \subseteq D$ .

Facts & Assumptions

Given: A pi-system $P$ on $X$ , its generated lambda-system $λ_{X} (P)$ , and an arbitrary lambda-system $D$ containing $P$ .

[L1]

If $P$ is a pi-system on $X$ , then $λ_{X} (P)$ is closed under binary intersections (The lambda-system generated by a pi-system is closed under finite intersections).

[L2]

A lambda-system on $X$ closed under binary intersections is a sigma-algebra on $X$ (A lambda-system closed under finite intersections is a sigma-algebra).

[L3]

The family $λ_{X} (P)$ is the smallest lambda-system containing $P$ (The generated lambda-system exists and is minimal).

[L4]

The family $σ_{X} (P)$ is the smallest sigma-algebra containing $P$ (Nonempty intersections of sigma-algebras are sigma-algebras, so the generated sigma-algebra exists and is minimal).

Proof

technique · direct

1.1

By [L1], [L2], and [L3], $λ_{X} (P)$ is a sigma-algebra containing $P$ . Hence [L4] gives $σ_{X} (P) \subseteq λ_{X} (P)$ .

L1L2L3L4

1.2

The sigma-algebra $σ_{X} (P)$ is a lambda-system: it contains $X$ , is closed under relative differences because it is closed under complements and intersections, and is closed under increasing countable unions. Since it contains $P$ , [L3] gives $λ_{X} (P) \subseteq σ_{X} (P)$ .

L3L4algebra

2.1

Steps 1.1 and 1.2 prove equality. Minimality in [L3] also gives $λ_{X} (P) \subseteq D$ , so $σ_{X} (P) \subseteq D$ .

step 1.1step 1.2L3∎

Depends on

Used by

Nothing in the library uses this result yet.

Dependency tree · next 3 levels

Direct dependencies and their dependencies through the next three levels: 11 results over 4 levels. An arrow runs from a result to what uses it, and this result sits at the bottom with a heavier outline. Click the chart to enlarge it.

Sources

A. Dembo, Probability Theory lecture notes, Theorem 1.1.38 (standard reference, not scraped)