proof of equivalent definitions of analytic sets for measurable spaces

Let $(X,\mathcal{F})$ be a measurable space and $A$ be a subset of $X$ . For any uncountable Polish space $Y$ with Borel $\sigma$ -algebra (http://planetmath.org/BorelSigmaAlgebra) $\mathcal{B}$ , we show that the following are equivalent.

1.

$A$ is $\mathcal{F}$ -analytic (http://planetmath.org/AnalyticSet2).
2.

$A$ is the projection (http://planetmath.org/GeneralizedCartesianProduct) of a set $S\in\mathcal{F}\otimes\mathcal{B}$ onto $X$ .

Here, $\mathcal{F}\otimes\mathcal{B}$ denotes the product $\sigma$ -algebra (http://planetmath.org/ProductSigmaAlgebra) of $\mathcal{F}$ and $\mathcal{B}$ .

(1) implies (2): Let $\mathcal{G}$ denote the paving consisting of the closed subsets of $Y$ . If $A$ is $\mathcal{F}$ -analytic then there exists a set $S\in(\mathcal{F}\times\mathcal{G})_{\sigma\delta}$ such that $A=\pi_{X}(S)$ , where $\pi_{X}\colon X\times Y\to X$ is the projection map (see proof of equivalent definitions of analytic sets for paved spaces). In particular, $\mathcal{G}\subseteq\mathcal{B}$ implies that $S$ is contained in the $\sigma$ -algebra $\mathcal{F}\otimes\mathcal{B}$ .

(2) implies (1): This is an immediate consequence of the result that projections of analytic sets are analytic.

Title	proof of equivalent definitions of analytic sets for measurable spaces
Canonical name	ProofOfEquivalentDefinitionsOfAnalyticSetsForMeasurableSpaces
Date of creation	2013-03-22 18:48:41
Last modified on	2013-03-22 18:48:41
Owner	gel (22282)
Last modified by	gel (22282)
Numerical id	4
Author	gel (22282)
Entry type	Proof
Classification	msc 28A05