countable unions and intersections of analytic sets are analytic
A property of analytic sets (http://planetmath.org/AnalyticSet2) which makes them particularly suited to applications in measure theory is that, in common with -algebras (http://planetmath.org/SigmaAlgebra), they are closed under countable unions and intersections.
Let be a nonempty paving on a set such that the complement (http://planetmath.org/Complement) of any is a union of countably many sets in .
Then, every set in the -algebra generated by is -analytic.
That the corollary does indeed follow from Theorem 1 is a simple application of the monotone class theorem. First, as the collection 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 (http://planetmath.org/RingOfSets). Then, Theorem 1 says that is closed under taking limits of increasing and decreasing sequences of sets. So, by the monotone class theorem, it contains the -algebra generated by .
|Title||countable unions and intersections of analytic sets are analytic|
|Date of creation||2013-03-22 18:45:18|
|Last modified on||2013-03-22 18:45:18|
|Last modified by||gel (22282)|