Nonempty intersections of sigma-algebras are sigma-algebras, so the generated sigma-algebra exists and is minimal

Depends on

• The sigma-algebra generated by a family of sets

Used by

• An unbounded predictable transform may lose integrability Counterexample
• Continuous-time filtrations and all-pairs martingales Definition
• Natural and usual augmented Brownian filtrations Definition
• Natural filtration of a process Definition
• Progressively measurable and predictable processes Definition
• The Borel sigma-algebra of a topological space Definition
• The product sigma-algebra and its finite iterates Definition
• A dyadic martingale converges to the original L1 variable Example
• Partial sums of independent centered variables are a martingale Example
• Polya urn proportion martingale Example
• Product martingale from independent mean one factors Example
• Approximation in symmetric difference by a generating algebra Lemma
• Borel sigma-algebra of continuous path space is generated by coordinates Lemma
• The sigma-algebra generated by the half-open boxes of ℝⁿ is the Borel sigma-algebra Lemma
• A continuous map has Borel preimages of Borel sets Theorem
• A countable partition generates exactly the unions of its blocks, and the resulting sigma-algebra is countable exactly for a finite partition Theorem
• A translation-invariant measure on the Borel sets of ℝⁿ giving the unit cube measure one is the restriction of Lebesgue measure Theorem
• An invertible linear map of ℝⁿ scales the Lebesgue measure of every Borel set by a positive constant depending only on the map Theorem
• Assuming countable choice, a generated sigma-algebra is obtained in omega-one stages of complements and countable unions Theorem
• Assuming countable choice, a premeasure extends through its induced outer measure Theorem
• Assuming countable choice, every Borel subset of ℝⁿ is Lebesgue measurable Theorem
• Cut locus of a point has riemannian volume zero Theorem
• Dynkin's pi-lambda theorem Theorem
• Every Borel set is Carathéodory measurable for a metric outer measure Theorem
• Every section of a product-measurable set is measurable Theorem
• Future-path Markov property Theorem
• Generated sigma-algebras are monotone in their generators and idempotent Theorem
• Generating a sigma-algebra commutes with taking traces Theorem
• The monotone class generated by an algebra equals the sigma-algebra it generates Theorem

Cited to discharge well-definedness by The sigma-algebra generated by a family of sets.

Dependency tree · two levels

2 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, Lemma 2.7 (standard reference, not scraped)
• T. Tao, An Introduction to Measure Theory, Exercises 1.4.13-1.4.14 (standard reference, not scraped)