Erdős problem and quadratic equation (Q533305)

The authors investigate an Erdős problem on almost quadratic functions on \(\mathbb R\). Erdős problem. Let \(f :\mathbb R \to\mathbb R\) be a function such that \(f(x+y) = f(x) + f(y)\) for almost all \((x,y)\in\mathbb R \times\mathbb R\). Does there exist an additive function \(F :\mathbb R \to\mathbb R\) such that \(f(x) = F(x)\) for almost all \(x \in\mathbb R\)?