On the methods of reduction of some types of Marczerwski-Burstin measurable functions to continuous functions on products of perfect sets

In this paper we provide product-wise generalizations of certain Marczewski-Burstin bases such as sets with (s)-property and completely Ramsey sets. For each of those we prove - as an analog of classical Luzin's and Eggleston's theorems that functions measurable with respect to those families can be reduced to continuous functions on a product of perfect sets. Moreover we also provide a way to reduce sequences of such functions to continuity, providing generalizations to Laver's extension of Halpern-Lauchli and Harrington theorems.