We are interested in the limit behaviour of Birkhoff sums over an infinite sigma-finite measure space. If the observable is integrable then — by a classical theorem by Aaronson — there exists no sequence of real numbers such that the Birkhoff sum normed by this sequence converges almost surely to 1. Under strong mixing conditions on the underlying system we prove a generalized strong law of large numbers for integrable observables using a truncated sum. We will see how this truncation works, how it is related to trimming in a finite system. For f not integrable we give conditions on f such that the Birkhoff sum normed by a sequence of real numbers converges almost surely to 1. This is joint work with Claudio Bonanno.