Borel hierarchies in infinite products of Polish spaces

Let H be a product of countably infinite number of copies of an uncountable Polish space X. Let ${\mathrm{\Sigma }}_{\xi }$ $({\bar{\mathrm{\Sigma }}}_{\xi })$ be the class of Borel sets of additive class \xi for the product of copies of the discrete topology on X (the Polish topology on X), and let ${\cal B} = \cup_{\xi < \omega_1} \bar{\Sigma}\xi$. We prove in the L'{e}vy–Solovay model that \bar{\Sigma}\xi =\Sigma_{\xi}\cap {\cal B} for $1\le \xi <{\omega }_1$.