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

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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.

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Acknowledgements

The author wishes to give his thanks to two anonymous referees for comments. The author was supported by JSPS KAKENHI 19K14549.

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  1. Kazuki Okamura

    Present address: Department of Mathematics, Faculty of Science, Shizuoka University, Shizuoka, Japan

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  1. School of General Education, Shinshu University, Matsumoto, Japan

    Kazuki Okamura

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  1. Kazuki Okamura
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Correspondence to Kazuki Okamura.

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Okamura, K. Non-existence of measurable solutions of certain functional equations via probabilistic approaches. Aequat. Math. 95, 629–637 (2021). https://doi.org/10.1007/s00010-021-00811-z

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  • DOI: https://doi.org/10.1007/s00010-021-00811-z

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