direct images of analytic sets are analytic
For measurable spaces and , consider a measurable function . By definition, the inverse image will be in whenever is in . However, the situation is more complicated for direct images (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 spaces is closed under direct images.
Let be a Borel measurable function between Polish spaces and . Then, the direct image is analytic whenever is an analytic subset of .
|Title||direct images of analytic sets are analytic|
|Date of creation||2013-03-22 18:46:33|
|Last modified on||2013-03-22 18:46:33|
|Last modified by||gel (22282)|