# direct images of analytic sets are analytic

For measurable spaces $(X,\mathcal{F})$ and $(Y,\mathcal{G})$, consider a measurable function $f\colon X\rightarrow Y$. By definition, the inverse image $f^{-1}(A)$ will be in $\mathcal{F}$ whenever $A$ is in $\mathcal{G}$. 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.

###### Theorem.

Let $f\colon X\rightarrow Y$ 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 DirectImagesOfAnalyticSetsAreAnalytic 2013-03-22 18:46:33 2013-03-22 18:46:33 gel (22282) gel (22282) 5 gel (22282) Theorem msc 28A05