On approximate group homomorphisms
\textit{I. Farah} [Combinatorica 20, No. 1, 47--60 (2000; Zbl 1008.28004)] investigated mappings between finite product groups which are approximately homomorphisms with respect to some metric on the range. After that \textit{B. Przebieracz} [Result. Math. 72, No. 4, 1959--1966 (2017; Zbl 1385.39007)] presented the versions of Farah's results without extra assumptions but \(G\) is assumed to be finite. In this paper, the authors prove the following statement: let \(G\) be a group (not necessarily finite) with a right-invariant, finitely additive probability measure \(\mu : P(G)\to [0,1]\), \(H\) be a group with an invariant metric \(d:H\times H \to [0, \infty), \varepsilon \geq 0\), and let \(0\leq \delta <\frac{1}{12}\). If a function \(f:G\to H\) satisfies \(\mu^2(\{(x,y):\,\,\,d(f(xy), f(x)f(y))>\varepsilon \}\leq \delta,\) then for every \(\zeta > 0\) there exists a map \(F:G\to H\) such that \(d(F(xy), F(x)F(y))\leq 24\varepsilon,\,\,\, x, y \in G\) and \(\mu (\{x\in G: d(f(x),F(x))>\varepsilon\})\leq 4\delta +\zeta.\)
scientific article; zbMATH DE number 6877138