countable unions and intersections of analytic sets are analytic

A property of analytic setsMathworldPlanetmath ( which makes them particularly suited to applications in measure theory is that, in common with σ-algebras (, they are closed under countableMathworldPlanetmath unions and intersectionsMathworldPlanetmathPlanetmath.

Theorem 1.

Let (X,F) be a paved space and (An)nN be a sequencePlanetmathPlanetmath of F-analytic sets. Then, nAn and nAn are F-analytic.

A consequence of this is that measurable setsMathworldPlanetmath are analytic, as follows.


Let F be a nonempty paving on a set X such that the complement ( of any SF is a union of countably many sets in F.

Then, every set A in the σ-algebra generated by F is F-analytic.

For example, every closed subset of a metric space X is a union of countably many open sets. Therefore, the corollary shows that all Borel sets are analytic with respect to the open subsets of X.

That the corollary does indeed follow from Theorem 1 is a simple application of the monotone class theorem. First, as the collectionMathworldPlanetmath a() of -analytic sets is closed under countable unions and finite intersections, it will contain all finite unions of finite intersections of sets in and their complements, which is an algebra ( Then, Theorem 1 says that a() is closed under taking limits of increasing and decreasing sequences of sets. So, by the monotone class theorem, it contains the σ-algebra generated by .

Title countable unions and intersections of analytic sets are analytic
Canonical name CountableUnionsAndIntersectionsOfAnalyticSetsAreAnalytic
Date of creation 2013-03-22 18:45:18
Last modified on 2013-03-22 18:45:18
Owner gel (22282)
Last modified by gel (22282)
Numerical id 4
Author gel (22282)
Entry type Theorem
Classification msc 28A05