Monte Carlo integration

this is perhaps more a comment than an answer, but here is one expert opinion on whether one can base the foundations of integration theory on Monte Carlo integration:

Another way to obtain continuous measure as the limit of discrete measure is via Monte Carlo integration, although in order to rigorously introduce the probability theory needed to set up Monte Carlo integration properly, one already needs to develop a large part of measure theory, so this perspective, while intuitive, is not suitable for foundational purposes.