Alphabeta MathDefinitionDefinition: Literature-sourcedProof: 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.
Algebras of subsets
Definition
Let be a set. An algebra of subsets of is a family such that:
- ;
- if , then ;
- if , then .
Thus an algebra is closed under complements relative to its fixed ambient set, finite unions, finite intersections, and differences. In particular belongs to every algebra on .
Used by
- Assuming countable choice, the outer set function induced by a premeasure is an outer measure Corollary
- Zero on finite sets and infinity on cofinite sets is finitely additive but not a premeasure Counterexample
- Binary-sequence cylinders and fair-coin content Definition
- Borel master codes for null and meagre sets Definition
- Premeasures on algebras of sets Definition
- Sigma-algebras Definition
- The algebra of finite disjoint unions of half-open intervals in ℝ with extended endpoints Definition
- FALSE: every finitely additive nonnegative function on an algebra extends to a measure False statement
- FALSE: every monotone class is an algebra False statement
- FALSE: the covering construction agrees with every finitely additive source function False statement
- A countable algebra cover disjointifies inside the covered algebra set Lemma
- A countable generator of a sigma-algebra yields a countable algebra of sets Lemma
- An algebra closed under countable disjoint unions is a sigma-algebra Lemma
- An algebra closed under increasing countable unions is a sigma-algebra Lemma
- Approximation in symmetric difference by a generating algebra Lemma
- Every member of the generated monotone class intersects every original algebra member inside the generated class Lemma
- Finite disjoint unions of measurable rectangles form an algebra generating the product sigma-algebra Lemma
- Finite-coordinate cylinder sets form an algebra Lemma
- Product rectangle kernels are dense in product L two Lemma
- The fair-coin measure on Cantor space Lemma
- The monotone class generated by an algebra is closed under complements Lemma
- Carathéodory measurable sets form an algebra Proposition
- The elementary sets form an algebra of subsets of ℝⁿ containing every half-open box Proposition
- A sigma-finite premeasure has at most one extension to its generated sigma-algebra Theorem
- Cut locus of a point has riemannian volume zero Theorem
Dependency tree · 0 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 2.1 (standard reference, not scraped)