  We prove a law of large numbers in terms of complete convergence of
independent random variables taking values in increments of monotone functions,
with convergence uniform both in the initial and the final time. The result
holds also for the random variables taking values in functions of $2$
parameters which share similar monotonicity properties as the increments of
monotone functions. The assumptions for the main result are the H\"older
continuity on the expectations as well as moment conditions, while the sample
functions may contain jumps. In particular, we can apply the results to point
processes (counting processes) which lack Markov or martingale type properties.
