Algebraic genericity within the class of sup-measurable functions

Abstract

It is well known that there exist real-valued, non-measurable functions on the plane, whose restrictions to the graph of all measurable functions on the real line are measurable. In this paper, it is shown that there is, in fact, a continuum-dimensional vector space all of whose members, except for zero, are functions on the plane satisfying the mentioned property. Versions of this fact are proved in the Baire category setting and in the case where the graphs are generated by softer functions.