Abstract: In this paper we use a filling scheme technique to prove the integrability of two counting functions. The integrability of one of these functions implies the a.e. convergence for the powers $T^nf(x)/n$ where T belongs to a certain class of nonlinear operators. The other counting function generalizes the upcrossing function considered by Bishop to the case of Chacon processes. In the last section we prove the connection between our results and previous results by Bishop. We also provide a result which connects upcrossings and oscillations.