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Abstract: 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 adding a suitable number of terms depending on the point of evaluation. 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. Finally, we look at the example of the backwards continued fractions where we have an observable which is non-integrable wrt the measure restrictet to a finite measure set. We give a proper truncation also for this system. This is joint work with Claudio Bonanno.