TOPICS
Search

Sigma-Algebra


Let X be a set. Then a sigma-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 sigma-algebra containing S, namely the power set of X. By taking the intersection of all sigma-algebras containing S, we obtain the smallest such sigma-algebra. We call the smallest sigma-algebra containing S the sigma-algebra generated by S.


See also

Borel Sigma-Algebra, Borel Space, Measurable Set, Measurable Space, Measure Algebra, Standard Space

Explore with Wolfram|Alpha

References

Jech, T. J. Set Theory, 2nd ed. Berlin: Springer-Verlag, p. 494, 1997.

Referenced on Wolfram|Alpha

Sigma-Algebra

Cite this as:

Weisstein, Eric W. "Sigma-Algebra." From MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/Sigma-Algebra.html

Subject classifications