Gw mau tau Conto dari Aljabar sigma. Jelasin dalam bahasa sunda

Mun S mangrupa sét naon baé, then the family consisting only of the empty set and S is a σ-algebra over S, the so-called trivial σ-algebra. Another σ-algebra over S is given by the full power set of S.

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

If U is an arbitrary family of subsets of S 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 S that contains U, namely the power set of S.

Let Φ be the family of all σ-algebras over S that contain U (that is, a σ-algebra 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 σ-algebras in Φ. σ(U) is then the smallest σ-algebra over S that contains U.

This léads 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 théory.