Sigma-algebra: Encyclopedia II - Sigma-algebra - Examples

Sigma-algebra - Examples

If X is any set, then the family consisting only of the empty set and X is a σ-algebra over X, the so-called trivial σ-algebra. Another σ-algebra over X is given by the full power set of X. The collection of subsets of X which are countable or whose complements are countable is a σ-algebra, which is distinct from the powerset of X iff X is uncountable.

If {Σ_a} is a family of σ-algebras over X, then the intersection of all Σ_a is also a σ-algebra over X.

If U is an arbitrary family of subsets of X then we can form a special σ-algebra from U, called the σ-algebra generated by U. We denote it by σ(U) and define it as follows. First note that there is a σ-algebra over X that contains U, namely the power set of X. Let Φ be the family of all σ-algebras over X that contain U (that is, a σ-algebra Σ over X is in Φ if and only if U is a subset of Σ.) Then we define σ(U) to be the intersection of all σ-algebras in Φ. σ(U) is then the smallest σ-algebra over X that contains U. For a simple example, consider the set X={1,2,3}. Then the σ-algebra generated by the subset {1} is σ({1}) = { ∅, {1}, {2,3}, X}. Note that by an abuse of notation, when my collection of subsets C is a singleton containing only A, one may write σ(A) instead of σ(C).

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

On the Euclidean space Rⁿ, another σ-algebra is of importance: that of all Lebesgue measurable sets. This σ-algebra contains more sets than the Borel algebra on Rⁿ and is preferred in integration theory.

Adapted from the Wikipedia article "Examples", under the G.N U Free Docmentation License. Please also see http://en.wikipedia.org/wiki