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 attention to a specific collection of subsets of $E$ , requiring that this collection be closed under operations that 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:

• $\emptyset\in\F$ .
• 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 sets that 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 $\powerset{E}$ is a $\sigma$ -algebra in $E$ , as is the set $\{\emptyset,E\}$ .

A more interesting example is the Borel $\sigma$ -algebra 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 $\R$ .