Subadditive Maps and Functional Equations

Given a group $G$ and a set $\Omega$, we say that a map $F\colon G\to 2^{\Omega}$ is subadditive if $F(gh) \subset F(g)\cup F(h)$ for all $g,h\in G$. Our main result on subadditive maps is that $|\bigcup_{g\in G}F(g)| \le 4 \sup_{g\in G}|F(g)|$, where $|M|$ denotes the number of elements of a subset $M\subset \Omega$. We also consider some extensions of this inequality to maps with values in the $\sigma$-algebra of all measurable subsets of a measure space and to maps with values in subspaces of a linear space. As an application, we obtain a description of solutions of some functional equations related to addition theorems.