Category analogue of sup-measurability problem (Q2706909)

A function \(F\colon{\mathbf R}^2\to{\mathbf R}\) is called Baire sup-measurable provided \(F_f\colon{\mathbf R}\to{\mathbf R}\) given by \(F_f(x)=F(x,f(x))\) has the Baire property for each function \(f\colon{\mathbf R}\to{\mathbf R}\) with the Baire property. A Baire sup-measurable function without the Baire property has been constructed under CH by \textit{E.~Grande} and \textit{Z.~Grande} [Fundam. Math. 121, 199-211 (1984; Zbl 0573.28007)]. In this paper the authors show that the existence of such functions cannot be proved in ZFC. This is a consequence of the following theorem: It is consistent with the set theory that for every \(A\subset 2^{\omega}\times 2^{\omega}\) for which the sets \(A\) and \(A^c\) are nowhere meager in \(2^{\omega}\times 2^{\omega}\) there exists a homeomorphism \(f\) from \(2^{\omega}\) onto \(2^{\omega}\) such that the set \(\text{pr}(A\cap f)\) does not have the Baire property in \(2^{\omega}\). This theorem is proved by the method of iterated forcing.