Alphabeta MathDefinitionDefinition: AI-adaptedProof: Not applicablejudge 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.
Pi-systems
Definition
Let be a set. A pi-system on is a nonempty family closed under binary intersections: if , then .
The nonempty-family requirement is the convention used here. It does not require and it does not add an empty-intersection axiom.
Used by
- Two four-point probability measures agree on a generating family that is not a pi-system Counterexample
- Closed left rays form a pi-system generating the Borel sigma-algebra on the real line Example
- FALSE: agreement on a generating pi-system always determines a signed measure False statement
- FALSE: agreement on an arbitrary generating family determines a measure False statement
- FALSE: every lambda-system is closed under finite intersections False statement
- Finite measures agreeing on a generating pi-system and on the whole space are equal Lemma
- Finite-coordinate cylinders form a π-system Lemma
- The lambda-system generated by a pi-system is closed under finite intersections Lemma
- A sigma-finite premeasure has at most one extension to its generated sigma-algebra Theorem
- A translation-invariant measure on the Borel sets of ℝⁿ giving the unit cube measure one is the restriction of Lebesgue measure Theorem
- Borel harmonicity and comparison of harmonic measure Theorem
- For sigma-finite factors, the product measure exists, has the rectangle formula, is sigma-finite, and is unique Theorem
- Independent pi-systems generate independent sigma-algebras Theorem
- Independent random elements are characterized by finite rectangle probabilities Theorem
- Measures agreeing on a generating pi-system are equal under an increasing finite-measure exhaustion from that pi-system Theorem
- On Borel subsets of Rᵐ⁺ⁿ, the product lambdaₘ times lambdaₙ agrees with lambdaₘ₊ₙ Theorem
- Polar coordinates decompose Lebesgue measure into rⁿ⁻¹ dr d sigma Theorem
Dependency tree · 0 levels
Nothing. This result depends on no other item in the library.
Sources
- A. Dembo, Probability Theory lecture notes, Definition 1.1.36 (standard reference, not scraped)