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

Let G be a group and Omega be an arbitrary set. A map F : G -> 2(Omega) is called subadditive if F(gh) subset of F(g) boolean OR F(h) for all g, h is an element of G. Denoting by vertical bar M vertical bar the number of elements of a subset M subset of Omega we show that vertical bar U-g is an element of G F(g)vertical bar <= 4 sup(g is an element of G) vertical bar F(g)vertical bar. 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. (C) 2012 Elsevier Inc. All rights reserved.