Under appropriate conditions on abelian topological groups G and H, an orthogonality ⟂⊂G2 and a σ-algebra ???? of subsets of G we prove that if at least one of the functions f,g,h: G→H satisfying
f(x+y) − g(x) − h(y) ∈ K for x,y∈G such that x⟂y,
where K is a discrete subgroup of H, is continuous at a point or ????-measurable, then there exist: a continuous additive function A: G→H, a continuous biadditive and symmetric function B: G×G→H and constants a,b∈H such that
f(x) − B(x,x) − A(x) − a ∈ K,
g(x) − B(x,x) − A(x) − b ∈ K,
h(x) − B(x,x) − A(x) − a + b ∈ K
for x∈G and
B(x,y) = 0 for x,y∈G such that x⟂y.
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Vol. 26 (2012)
Published: 2013-01-15
10.1515/amsil
English