Stability of a conditional Cauchy equation on a set of measure zero

Abstract

In this paper, we prove the Hyers–Ulam stability theorem when ${f, g, h : \mathbb{R} \to \mathbb{R}}$ satisfy

$$|f(x + y) - g(x) - h(y)| \leq \epsilon$$

in a set ${\Gamma \subset \mathbb{R}^{2}}$ of measure ${m(\Gamma) = 0}$, which refines a previous result in Chung (Aequat Math 83:313–320, 2012) and gives an affirmative answer to the question in the paper. As a direct consequence we obtain that if ${f, g, h : \mathbb{R} \to \mathbb{R}}$ satisfy the Pexider equation

$$f(x + y) - g(x) - h(y) = 0$$

in ${\Gamma}$, then the equation holds for all ${x, y \in \mathbb{R}}$. Using our method of construction of the set, we can find a set ${\Gamma \subset \mathbb{R}^{2n}}$ of 2n-dimensional measure 0 and obtain the above result for the functions ${f, g, h : \mathbb{R}^{n} \to \mathbb{C}}$.