Sigma-algebras

Depends on

• Algebras of subsets

Used by

• No sigma-algebra is countably infinite Corollary
• A set can have measurable horizontal and vertical sections and still fail to be product-measurable Counterexample
• The zero-countable / infinity-cocountable measure space breaks the p=1 endpoint of duality Counterexample
• A complex measure is a finite-valued countably additive set function Definition
• A signed measure is countably additive and takes at most one infinite value Definition
• Complex simple functions as finite sums of measurable indicators Definition
• Filtration and filtered probability space Definition
• Finitely additive nonnegative set functions Definition
• Independent sigma-algebras and independent events Definition
• Measurable spaces and measurable sets Definition
• Measures on sigma-algebras Definition
• Projection valued measure Definition
• Strict and mod-null invariant sigma-algebras Definition
• The Dirac set function at a point 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
• Assuming countable choice, zero on countable sets and infinity on cocountable sets is a non-semifinite measure Example
• The trivial and discrete sigma-algebras are the two extremes Example
• FALSE: a measure on an infinite set that vanishes on every singleton is the zero measure False statement
• FALSE: every finitely additive nonnegative function on an algebra extends to a measure False statement
• FALSE: every subset of the real line is Borel False statement
• FALSE: if every horizontal and vertical section is measurable, then the set is product-measurable 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 shear sends the unit cube to a set of Lebesgue measure one 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
• Assuming countable choice, the completion domain is a sigma-algebra Lemma
• Elementary bounds on ideal cardinal invariants Lemma
• Equivalent event tests for a discrete stopping time Lemma
• The sigma-algebra generated by the half-open boxes of ℝⁿ is the Borel sigma-algebra Lemma
• The stopping-time sigma-algebra is a sigma-algebra Lemma
• A box with a degenerate side is Lebesgue null, and so is every coordinate hyperplane in ℝⁿ Proposition
• Closure properties of measurable functions used by the integral Proposition
• Every at most countable subset of ℝⁿ is Lebesgue null; in particular λ₁(ℚ)=0 Proposition
• 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

…and 7 more results.

Dependency tree · one level

1 result within one dependency step 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, Definition 2.1 (standard reference, not scraped)
• T. Tao, An Introduction to Measure Theory, Definition 1.4.12 (standard reference, not scraped)