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

TL;DR

This paper proves that certain functional equations involving measurable functions, related to distributions like uniform, Cauchy, and arctan, have no measurable solutions, using probabilistic methods.

Contribution

It demonstrates the non-existence of measurable solutions for specific functional equations linked to distribution characterizations, employing probabilistic techniques.

Findings

No measurable solutions exist for equations associated with uniform or Cauchy distributions.

The proof uses characterization of the Dirac measure and applies to the arctan equation.

Results highlight limitations of measurable solutions in distribution-related functional equations.

Abstract

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