Math 2.0

definition

Sigma-algebra

Definition

Let Ω\Omega be a set. A σ-algebra on Ω\Omega is a subset F\mathcal{F} of the power set of Ω\Omega with these three properties. The set Ω\Omega is in F\mathcal{F}. If A∈FA \in \mathcal{F}, then its complement Ω∖A\Omega \setminus A is in F\mathcal{F}. If A1,A2,A3,…A_1, A_2, A_3, \dots is a sequence of elements of F\mathcal{F}, then the union ⋃n=1∞An\bigcup_{n=1}^{\infty} A_n is in F\mathcal{F}. The pair (Ω,F)(\Omega, \mathcal{F}) is a measurable space, and the elements of F\mathcal{F} are the measurable sets.

Remarks

A σ-algebra contains ∅\varnothing and is closed under countable intersections, by De Morgan's laws. The power set of Ω\Omega and the family {∅,Ω}\{\varnothing, \Omega\} are σ-algebras on Ω\Omega. They are the largest and the smallest σ-algebras on Ω\Omega. For each family C\mathcal{C} of subsets of Ω\Omega, the intersection of all σ-algebras that contain C\mathcal{C} is a σ-algebra. It is the σ-algebra σ(C)\sigma(\mathcal{C}) generated by C\mathcal{C}. The Borel σ-algebra on R\mathbb{R} is the σ-algebra generated by the open intervals.

Explanations

L1 human

A σ-algebra is the list of the sets that we are allowed to measure. If we can measure a set, then we can measure its complement. If we can measure each set of a sequence, then we can measure their union. For a finite sample space we can usually measure every subset, so the σ-algebra is the power set.

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Sources

  1. Web https://en.wikipedia.org/wiki/Σ-algebra, Definition

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