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We present some extension of the concept of an invariant mean to a space of vector-valued mappings defined on a semigroup. Next, we apply it to the study of the stability of some functional equation.
In [4], assuming among others subadditivity and submultiplicavity of a function ψ: [0, ∞)→[0, ∞), the authors proved a Hyers-Ulam type stability theorem for “ψ-additive” mappings of a normed space into a normed space. In this note we show that the assumed conditions of the function ψ imply that ψ=0 and, consequently, every “ψ-additive” mapping must be additive
We study and solve several functional equations which yield necessary and sufficient conditions for the sum of two uniformly distributed random variables to be uniformly distributed.
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