Proper families and almost additive functions

Summary.

We generalize the well-known result of R. Ger (see [6]) on almost additive functions, i.e. we prove the following¶¶Theorem. Let S be a left reversible semigroup and let $ {\cal I} $ be a left translation invariant family of subsets of S which is 4-proper, that is such that¶$\forall A_1,\ldots,A_4 \in {\cal I} : A_1 \cup \ldots \cup A_4 \neq S$.¶Let H be a group and let $f : S \to H$ satisfy¶¶$f(x+y) = f(x)+f(y)$ for $\Omega({\cal I})-a.a.\,(x,y) \in S \times S$.¶Then there exists a unique additive function $F : S \to H$ such that¶¶$f(x)=F(x)$ for ${\cal I}-a.a.\,x \in S$.¶Moreover, we show that the assumption that ${\cal I}$ is 4-proper is essential.