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

TL;DR

This paper generalizes classical theorems to show that certain measurable functions can be reduced to continuous functions on products of perfect sets, extending the scope of reduction methods in descriptive set theory.

Contribution

It introduces product-wise generalizations of Marczewski-Burstin bases and establishes new reduction theorems for measurable functions on these families.

Findings

Functions measurable with respect to these families can be reduced to continuous functions on product of perfect sets.

Established analogs of Luzin and Eggleston theorems for these generalized families.

Provided a method to reduce sequences of such functions to continuity, extending Laver's work.

Abstract

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-L\"auchli and Harrington theorems.