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 ${\pi}_{X}:X\times Y\to X$ 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  20130322 18:48:41 
Last modified on  20130322 18:48:41 
Owner  gel (22282) 
Last modified by  gel (22282) 
Numerical id  4 
Author  gel (22282) 
Entry type  Proof 
Classification  msc 28A05 