sigma-algebra

A sigma-algebra is the bookkeeping device that decides which sets are allowed to be measured. You cannot consistently assign a length to every subset of the real line — some sets are too wild — so you choose a tidy family of “admissible” sets in advance, and require only that this family be closed under the operations you actually use: complementing a set and forming countable unions.

Precisely, a collection A of subsets of a set X is a sigma-algebra on X if it contains X itself, is closed under complementation (A in A implies X minus A in A), and is closed under countable unions (if A_1, A_2, ... are all in A, then their union is in A). From these axioms it follows that A also contains the empty set and is closed under countable intersections (by De Morgan’s laws) and under set differences.

The prefix “sigma” signals the countable, not merely finite, closure — this is what makes limits behave. Many natural sigma-algebras are described not by listing their members but by the smallest one containing some generating family: the Borel sigma-algebra is generated by the open sets. There is always a smallest sigma-algebra containing any given family, because an arbitrary intersection of sigma-algebras is again a sigma-algebra.