Category analogue of sup-measurability problem

A function F : ℝ² → ℝ is called sup-measurable if F_f : ℝ → ℝ given by F_f (x) = F(x, f(x)), x ∈ ℝ, is measurable for each measurable function f : ℝ → ℝ. It is known that under different set theoretical assumptions, including CH, there are sup-measurable non-measurable functions, as well as their category analogues. In this paper we will show that the existence of the category analogues of supmeasurable non-measurable functions is independent of ZFC. A similar result for the original measurable case is the subject of a work in prepartion by Ros lanowski and Shelah.