Asymptotic properties of the process counted with a random characteristic in the context of fragmentation processes

In this paper we prove a strong law of large numbers and its L^1-convergence counterpart for the process counted with a random characteristic in the context of self-similar fragmentation processes. This result extends a somewhat analogical result by Nerman for general branching processes to fragmentation processes. In addition, we apply the general result of this paper to a specific example that in particular extends a limit theorem, concerning the fragmentation energy, by Bertoin and Mart\'inez from L^1-convergence to almost sure convergence. Our approach treats fragmentation processes with an infinite dislocation measure directly, without using a discretisation method. Moreover, we obtain a result regarding the asymptotic behaviour of the empirical mean associated with some stopped fragmentation process.