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 (http://planetmath.org/DirectImage), 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.

Theorem.

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