direct images of analytic sets are analytic

For measurable spacesMathworldPlanetmathPlanetmath (X,) and (Y,𝒢), consider a measurable functionMathworldPlanetmath f:XY. By definition, the inverse image f-1(A) will be in whenever A is in 𝒢. However, the situation is more complicated for direct imagesPlanetmathPlanetmath (, which in general do not preserve measurability. However, as stated by the following theorem, the class of analytic subsets of Polish spacesMathworldPlanetmath is closed under direct images.


Let f:XY be a Borel measurable function between Polish spaces X and Y. Then, the direct image f(A) is analytic whenever A is an analytic subset of X.

Title direct images of analytic sets are analytic
Canonical name DirectImagesOfAnalyticSetsAreAnalytic
Date of creation 2013-03-22 18:46:33
Last modified on 2013-03-22 18:46:33
Owner gel (22282)
Last modified by gel (22282)
Numerical id 5
Author gel (22282)
Entry type Theorem
Classification msc 28A05