Sigma-algebra

Builds on Set Power set Complement Union

Definition

Let $\Omega$ be a set. A σ-algebra on $\Omega$ is a subset $\mathcal{F}$ of the power set of $\Omega$ with these three properties. The set $\Omega$ is in $\mathcal{F}$ . If $A \in \mathcal{F}$ , then its complement $\Omega \setminus A$ is in $\mathcal{F}$ . If $A_1, A_2, A_3, \dots$ is a sequence of elements of $\mathcal{F}$ , then the union $\bigcup_{n=1}^{\infty} A_n$ is in $\mathcal{F}$ . The pair $(\Omega, \mathcal{F})$ is a measurable space, and the elements of $\mathcal{F}$ 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 $\Omega$ and the family $\{\varnothing, \Omega\}$ are σ-algebras on $\Omega$ . They are the largest and the smallest σ-algebras on $\Omega$ . For each family $\mathcal{C}$ of subsets of $\Omega$ , the intersection of all σ-algebras that contain $\mathcal{C}$ is a σ-algebra. It is the σ-algebra $\sigma(\mathcal{C})$ generated by $\mathcal{C}$ . The Borel σ-algebra on $\mathbb{R}$ is the σ-algebra generated by the open intervals.

Where this leads

386 pages build on this one.

Tag	`04WSW8T3SY`
URI	`m2://04WSW8T3SY`
Kind	definition
Writing	W1 stated
MSC	28A05
Defines	Sigma-algebra
Uses	Set, Power set, Complement, Union

Sigma-algebra

Definition

Remarks

Reading path

Where this leads

Further reading

What this page needs for W2

What links here

Used by