Sigma algebra

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A sigma algebra X over a set E is a family of subsets of E with the following properties:

1) E is in X,

2) If S is in X then so is the complement of X.

3) If S1, S2, S3, ... is a sequence in X then their countable union must also be in X.

Examples: If E is any set, then each of these two families is a sigma algebra: {0,E} (the trivial sigma algebra), {S;S is a subset of E} (the power set). If {Xa} is a family of sigma algebrae, then the intersection of all Xa is also a sigma algebra.

If F is an arbitrary family of sets in E then we can form a special sigma algebra from F, called the sigma algebra generated by F. We denote it by σ(F) and define it as follows. First note that there is a sigma algebra that contains F, namely the power set. Let G be the family of sigma algebrae containing F (that is, a sigma algebra X over E is in G if and only if F is a subset of X.) Then we simply define σ(F) to be the intersection of all sigma algebrae in G.

This leads to the most important example: the Borel sigma algebra in any topological space is the sigma algebra generated by the open sets (or, equivalently, by the closed sets.) It is important to note that this sigma algebra is not, in general, the power set. For a non-trivial example, see the vitali set.