Abstract
We show that, under suitable assumptions, a function f from a group (G, +) into a real or complex inner product space \({(H, (\cdot \vert \cdot))}\), satisfying the Fischer–Muszély functional equation
for all pairs (x, y) off a sufficiently small (negligible) subset of G 2 has to be almost everywhere equal to an additive map from G into H, i.e. the set \({\{x \in G: f(x) \neq a(x)\}}\) is small (negligible) in G. Small sets in G and G 2 are defined in an axiomatic way. Several corollaries illustrating some consequences of this result are presented.
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Dedicated to Professor János Aczél on his 90th birthday
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Ger, R. Fischer–Muszély functional equation almost everywhere. Aequat. Math. 89, 207–214 (2015). https://doi.org/10.1007/s00010-014-0299-x
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DOI: https://doi.org/10.1007/s00010-014-0299-x