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.
The monotone class generated by an algebra equals the sigma-algebra it generates
Statement
For every algebra of subsets of ,
Facts & Assumptions
Given: An algebra on and .
The family is the smallest monotone class containing (The generated monotone class exists and is minimal).
The family is closed under complements (The monotone class generated by an algebra is closed under complements) and binary intersections (The monotone class generated by an algebra is closed under finite intersections).
An algebra closed under increasing countable unions is a sigma-algebra (An algebra closed under increasing countable unions is a sigma-algebra).
The family is the smallest sigma-algebra containing (Nonempty intersections of sigma-algebras are sigma-algebras, so the generated sigma-algebra exists and is minimal).
Proof
By [L2], contains and is closed under complements and binary intersections, hence under finite unions; it is therefore an algebra.
Every sigma-algebra is closed under increasing unions and, by De Morgan's law, decreasing intersections. Thus is a monotone class containing , and minimality in [L1] gives .
Since is a monotone class by [L1], it is closed under increasing countable unions. Step 1.1 and [L3] make it a sigma-algebra containing , so [L4] gives .
The inclusions of steps 2.1 and 1.2 prove the equality.
Depends on
- Nonempty intersections of sigma-algebras are sigma-algebras, so the generated sigma-algebra exists and is minimal
- The generated monotone class exists and is minimal
- The monotone class generated by an algebra is closed under complements
- The monotone class generated by an algebra is closed under finite intersections
- An algebra closed under increasing countable unions is a sigma-algebra
Used by
- Bounded-function form of the Markov property Lemma
- Conditional-independence equivalences and preservation Lemma
- For sigma-finite measures, the section-measure functions are measurable Proposition
- Change of variables for an increasing absolutely continuous function Theorem
- For sigma-finite measures, the two section-measure integrals of a measurable set agree Theorem
- Ionescu-Tulcea construction of a Markov chain Theorem
- Kolmogorov zero-one law Theorem
- Lebesgue-Stieltjes measures on ℝ are outer regular and inner regular by compact sets Theorem
- Levy upward convergence of conditional expectations Theorem
- Markov property for bounded future path functionals Theorem
Dependency tree · two levels
10 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
- R. F. Bass, Real Analysis for Graduate Students, version 5.0, Theorem 2.10 (standard reference, not scraped)
- A. Dembo, Probability Theory lecture notes, Theorem 1.1.44 (standard reference, not scraped)