Subadditive set-functions on semigroups, applications to group representations and functional equations

Let G be a group and Ω be an arbitrary set. A map F :G → 2Ω is called subadditive if F(gh) ⊂ F(g) ∪ F(h) for all g,h ∈ G. Denoting by |M| the number of elements of a subset M ⊂ Ω we show that | g∈G F(g)| 4supg∈G |F(g)|. We also establish the extensions of this inequality to maps with values in measurable subsets of a measure space and to maps with values in subspaces of a linear space. We apply this technique to study some functional equations of addition theorem type. © 2012 Elsevier Inc. All rights reserved