Formally, X is a σalgebra if and only if it has the following properties:
From 1 and 2 it follows that S is in X; from 2 and 3 it follows that the σalgebra is also closed under countable intersections.
An ordered pair (S, X), where S is a set and X is a σalgebra over S, is called a measurable space.
If S is any set, then the family consisting only of the empty set and S is a σalgebra over S, the socalled trivial σalgebra. Another σalgebra over S is given by the full power set of S.
If {X_{a}} is a family of σalgebras over S, then the intersection of all X_{a} is also a σalgebra over S.
If U is an arbitrary family of subsets of S then we can form a special σalgebra from U, called the σalgebra generated by U. We denote it by σ(U) and define it as follows. First note that there is a σalgebra over S that contains U, namely the power set of S. Let Φ be the family of all σalgebras over S that contain U (that is, a σalgebra X over S is in Φ if and only if U is a subset of X.) Then we define σ(U) to be the intersection of all σalgebras in Φ. σ(U) is then the smallest σalgebra over S that contains U; its elements are all sets that can be gotten from sets in U by applying a countable sequence of the set operations union, intersection and complement.
This leads to the most important example: the Borel algebra over any topological space is the σalgebra generated by the open sets (or, equivalently, by the closed sets). Note that this σalgebra is not, in general, the whole power set. For a nontrivial example, see the Vitali set[?].
On the Euclidean space R^{n}, another σalgebra is of importance: that of all Lebesgue measurable sets. This σalgebra contains more sets than the Borel algebra on R^{n} and is preferred in integration theory.
See also measurable function.
Search Encyclopedia
