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 $X$ .

单词	DirectImagesOfAnalyticSetsAreAnalytic
释义	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 $X$ .
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