# Product of limit $\sigma$-algebras

Let $X$ and $Y$ be Polish (i.e. Borel subsets of separable completely metrizable) spaces. For a Polish space $Z$, let $\mathscr{S}(Z)$ denote the limit $\sigma$-algebra on $Z$, i.e. the smallest $\sigma$-algebra on $Z$ that contains the Borel subsets of $Z$ and is closed under the Suslin operation.

Question: Does $\mathscr{S}(X) \otimes \mathscr{S}(Y) = \mathscr{S}(X \times Y)$?

Every member of your limit $\sigma$-algebra is both Lebesgue measurable and has the Baire property (for a proof, see section 29.B in Kechris book). A result of Mansfield and Rao implies that the universal analytic set in plane is not in the sigma algebra generated by rectangles with measurable (resp. Baire property) sides - For a short proof see Theorem 1 in this paper of Arnold Miller.