Alphabeta MathHow statement and proof provenance work
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The sigma-algebra generated by a family of sets
Definition
Let be a set and let . Put
The sigma-algebra generated by is
When the ambient set is clear, write . The existence and minimality implicit in this terminology are proved in Nonempty intersections of sigma-algebras are sigma-algebras, so the generated sigma-algebra exists and is minimal ↗.
Depends on
Used by
- The Borel sigma-algebra of a topological space Definition
- Assuming countable choice, a generated sigma-algebra is obtained in omega-one stages of complements and countable unions Theorem
- Nonempty intersections of sigma-algebras are sigma-algebras, so the generated sigma-algebra exists and is minimal Theorem
Dependency tree · next 3 levels
Direct dependencies and their dependencies through the next three levels: 2 results over 2 levels. 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, Lemma 2.7 (standard reference, not scraped)
- T. Tao, An Introduction to Measure Theory, Definition 1.4.14 (standard reference, not scraped)