  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.
