Let X be a set. Then a σ-algebra F is a nonempty collection of subsets of X such that the following hold: 1.X is in F. 2. If A is in F, then so is the complement of A. 3. If A_n is a sequence of elements of F, then the union of the A_ns is in F. If S is any collection of subsets of X, then we can always find a σ-algebra containing S, namely the power set of X. By taking the intersection of all σ-algebras containing S, we obtain the smallest such σ-algebra. We call the smallest σ-algebra containing S the σ-algebra generated by S.

Borel sigma-algebra | Borel space | measurable set | measurable space | measure algebra | standard space