On lineability of families of non-measurable functions of two variable

A function $$F:\mathbb {R}^2\rightarrow \mathbb {R}$$ is sup-measurable if, for each (Lebesgue) measurable function $$f:\mathbb {R}\rightarrow \mathbb {R}$$ , the Caratheodory superposition $$F_f:\mathbb {R}\rightarrow \mathbb {R}$$ given by $$F_f: x\mapsto F(x,f(x))$$ is measurable. The existence of non-measurable sup-measurable functions is independent of ZFC. We prove, assuming CH, that the family of all non-measurable sup-measurable functions $$F:\mathbb {R}^2\rightarrow \mathbb {R}$$ (plus the zero function) contains a linear vector space of dimension $$2^\mathfrak {c}$$ . A function $$F:\mathbb {R}^2\rightarrow \mathbb {R}$$ is separately measurable if all its vertical and horizontal sections are measurable. In the second part of this note we show that the family of non-measurable separately measurable functions is $$2^\mathfrak {c}$$ -lineable.