On lineability of families of non-measurable functions of two variable
Given a set \(X\) and a class~\(C\) of real-valued functions on~\(X\) one says that \(C\)~is \(\kappa\)-lineable (where \(\kappa\) is some cardinal) if \(C\cup\{0\}\) contains a \(\kappa\)-dimensional vector subspace of~\(\mathbb{R}^X\). Many classes are lineable simply because they are vector spaces (continuous, integrable, differentiable); hence lineability becomes non-trivial\slash interesting for classes that are non automatically closed under linear combinations. The author considers the class of functions on~\(\mathbb{R}^2\) that are sup-measurable: \(F:\mathbb{R}^2\to\mathbb{R}\) is sup-measurable if \(x\mapsto F(x,f(x))\) is measurable for every measurable function~\(f\) (`sup' is short for `superposition'). The Continuum Hypothesis implies that the class of sup-measurable but not measurable functions is \(2^\mathfrak{c}\)-lineable. A similar result obtains in ZFC for functions that are not measurable but all of whose vertical and horizontal sections are measurable.\par The paper contains many more results of this nature.
scientific article; zbMATH DE number 7352841
scientific article