Alphabeta MathHow 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.
Sigma-algebras
Definition
Let be a set. A sigma-algebra on is an algebra of subsets (Algebras of subsets) that is closed under countable unions: whenever is a sequence in ,
The pair then has a fixed ambient set . Complements in the sigma-algebra axioms always mean complements relative to that .
Depends on
Used by
- No sigma-algebra is countably infinite Corollary
- Measurable spaces and measurable sets Definition
- The sigma-algebra generated by a family of sets Definition
- The trace of a sigma-algebra on a subset Definition
- Assuming countable choice, the countable-cocountable family is a sigma-algebra Example
- The trivial and discrete sigma-algebras are the two extremes Example
- FALSE: every subset of the real line is Borel False statement
- FALSE: the union of an increasing sequence of sigma-algebras is a sigma-algebra False statement
- FALSE: the union of two sigma-algebras on one set is a sigma-algebra False statement
- A lambda-system closed under finite intersections is a sigma-algebra Lemma
- A sigma-algebra with a listed infinite subfamily contains a disjoint sequence of nonempty members 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
- Every sigma-algebra is a lambda-system and a monotone class Proposition
- A countable partition generates exactly the unions of its blocks, and the resulting sigma-algebra is countable exactly for a finite partition Theorem
- Assuming countable choice, every infinite sigma-algebra contains a copy of the power set of the natural numbers Theorem
- Sigma-algebras are closed under countable intersections, differences, symmetric differences, and set limits Theorem
Dependency tree · next 3 levels
Direct dependencies and their dependencies through the next three levels: 1 result over 1 level. 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
- R. F. Bass, Real Analysis for Graduate Students, version 5.0, Definition 2.1 (standard reference, not scraped)
- T. Tao, An Introduction to Measure Theory, Definition 1.4.12 (standard reference, not scraped)