Introduction

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

Definition

Given a set , a -algebra in is a collection of subsets of such that:

• .
• Any union of countably many elements of is an element of .
• The complement of any element of in is an element of .

Notes

It follows from the definition that any -algebra in also satisfies the properties:

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

Note that a -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 of subsets of , the -algebra generated by is defined to be the smallest -algebra in such that . This is well-defined, as the intersection of any non-empty collection of -algebras in is also a -algebra in .

Examples

For any set , the power set is a -algebra in , as is the set .

A more interesting example is the Borel -algebra in , which is the -algebra generated by the open subsets of , or, equivalently, the -algebra generated by the compact subsets of .