$\sigma$-algebra

Introduction

When defining a measure for a set $E$we usually cannot hope to make every subset of $E$ measurable.Instead we must usually restrict our attentionto a specific collection of subsets of $E$,requiring that this collection be closed under operationsthat we would expect to preserve measurability.A $\sigma$-algebra is such a collection.

Definition

Given a set $E$, a $\sigma$-algebra in $E$is a collection $ℱ$ of subsets of $E$ such that:

• $\mathrm{\varnothing }\in ℱ$.

• Any union of countably many elements of $ℱ$is an element of $ℱ$.

• The complement of any element of $ℱ$ in $E$is an element of $ℱ$.

Notes

It follows from the definition that any $\sigma$-algebra $ℱ$ in $E$also satisfies the properties:

• $E\in ℱ$.

• Any intersection of countably many elements of $ℱ$is an element of $ℱ$.

Note that a $\sigma$-algebra is a field of setsthat is closed under countable unions and countable intersections(rather than just finite unions and finite intersections).

Given any collection $C$ of subsets of $E$,the $\sigma$-algebra $\sigma (C)$ generated by $C$is defined to be the smallest $\sigma$-algebra in $E$such that $C\subseteq \sigma (C)$.This is well-defined,as the intersection of any non-empty collection of $\sigma$-algebras in $E$is also a $\sigma$-algebra in $E$.

Examples

For any set $E$,the power set $𝒫(E)$ is a $\sigma$-algebra in $E$,as is the set $\{\mathrm{\varnothing },E\}$.

A more interesting example is theBorel $\sigma$-algebra (http://planetmath.org/BorelSigmaAlgebra) in $ℝ$,which is the $\sigma$-algebra generated by the open subsets of $ℝ$,or, equivalently,the $\sigma$-algebra generated by the compact subsets of $ℝ$.