A function is called sup-measurable if Fƒ : ℝ → ℝ given by Fƒ(x) = F(x, ƒ(x)), x ∈ ℝ, is measurable for each measurable function ƒ : 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 sup-measurable 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.
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