Abstract
Let (X, +, μ) be a measurable group such that μ is complete and μ(X) = ∞, and let (E, +) be a metric group. Let f: X → E be any mapping. We prove that if there exists a p > 0 such that the function \((d(f(x + y), f(x) + f(y)))^p \) is majorizable by an integrable function then f is almost everywhere additive. Similar results we also obtain for the Jensen and Pexider equations.
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Tabor, J. Stability of the Cauchy type equations in \( {\cal L}_p \) norms. Results. Math. 32, 145–158 (1997). https://doi.org/10.1007/BF03322534
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DOI: https://doi.org/10.1007/BF03322534