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∈GF(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.