The sigma-algebra generated by a family of sets

Depends on

• Sigma-algebras

Used by

• Two four-point probability measures agree on a generating family that is not a pi-system Counterexample
• Borel master codes for null and meagre sets Definition
• Coordinate maps, finite-coordinate cylinders, and the cylinder σ-algebra Definition
• Tail sigma-algebra of a sequence Definition
• The Borel sigma-algebra of a topological space Definition
• The product sigma-algebra and its finite iterates Definition
• FALSE: agreement on an arbitrary generating family determines a measure False statement
• A countable generator of a sigma-algebra yields a countable algebra of sets Lemma
• Approximation in symmetric difference by a generating algebra Lemma
• Assuming countable choice, cylinder-measurable events depend on only countably many coordinates Lemma
• Borel sigma-algebra of continuous path space is generated by coordinates Lemma
• Finite measures agreeing on a generating pi-system and on the whole space are equal Lemma
• Finite-measure sets are approximable in measure by sets from a countable generating algebra Lemma
• The fair-coin measure on Cantor space Lemma
• The sigma-algebra generated by the half-open boxes of ℝⁿ is the Borel sigma-algebra Lemma
• Under sigma-finiteness, every Carathéodory measurable set differs from a generated measurable hull by a null set Lemma
• A sigma-finite premeasure has at most one extension to its generated sigma-algebra 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, a premeasure-induced outer measure is regular with generated measurable hulls Theorem
• Assuming countable choice, every Borel subset of ℝⁿ is Lebesgue measurable Theorem
• Cut locus of a point has riemannian volume zero Theorem
• Kolmogorov zero-one law Theorem
• Nonempty intersections of sigma-algebras are sigma-algebras, so the generated sigma-algebra exists and is minimal Theorem

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, Definition 1.4.14 (standard reference, not scraped)