In this paper, we introduce product-wise generalizations of certain Marczewski-Burstin bases, including sets with the (s)-property and completely Ramsey sets. For each of these families, we establish analogs of the classical Luzin and Eggleston theorems, showing that functions measurable with respect to these families can be reduced to continuous functions on products of perfect sets. Furthermore, we provide a method for reducilng sequences of such functions to continuity, which allows us to generalize Laver's extension of Halpern-Läuchli and Harrington theorems.