Non-existence of measurable solutions of certain functional equations via probabilistic approaches

This paper deals with functional equations in the form of $$f(x) + g(y) = h(x,y)$$ where h is given and f g are unknown. We will show that if a Borel measurable function associated characterizations uniform or Cauchy distributions, then there no solutions equation. Our proof uses characterization Dirac measure it also applicable to arctan