# 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. 1.

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

2. 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 ProofOfEquivalentDefinitionsOfAnalyticSetsForMeasurableSpaces 2013-03-22 18:48:41 2013-03-22 18:48:41 gel (22282) gel (22282) 4 gel (22282) Proof msc 28A05