On strong algebrability of families of non-measurable functions of two variables

TL;DR

This paper advances the understanding of the algebraic structure of non-measurable functions of two variables by demonstrating the existence of large free algebras within these classes, extending previous lineability results.

Contribution

It improves upon prior work by showing that most classes of non-measurable functions contain free algebras with continuum many generators.

Findings

Most classes contain free algebras with 2^c generators

Enhancement of lineability results to algebrability

Extension of previous results by Natkaniec

Abstract

Recently Tomasz Natkaniec in [On lineability of families of non-measurable functions of two variable. Rev. R. Acad. Cienc. Exactas F\'is. Nat. Ser. A Mat. RACSAM, 115(1):Paper No. 33, 10, 2021] studied the lineability problem for several classes of non-measurable functions in two variables. In this note we improve his results in the direction of algebrability. In particular, we show that most of the classes considered by Natkaniec contain free algebras with $2^{\mathfrak{c}}$ many generators.