Small subset of the plane which almost contains almost all Borel functions (Q1175105)

A \(G_{\delta}\) set \(B\subset\mathbb{R}\times\mathbb{R}\) is proved to exist such that all its vertical sections are of Lebesgue measure zero and for each Borel function \(f: \mathbb{R}\to\mathbb{R}\) the set \(\{x\in\mathbb{R};(x,f(x)+y)\in B\}\) is comeager for all but countably many \(y\in\mathbb{R}\). The construction of \(B\) is based on decompositions of \([0,1]\) to sums (modulo 1) of a finite set and of a finite union \(C\) of rational intervals with \(\lambda(C)\) small. The set \(B\) is used to show a simpler proof of a previous result of J. Cichoń and the author. Some properties of reals are derived by adding a Cohen real to the universe. The roles of measure and category in the main result cannot be interchanged.