An informal perspective on some implication of Monte Carlo on integral:<p>One of the most intuitive (and used in applications) definition of being integrable is Riemann integral based on the geometric idea that you can compute the area/volume by dividing region into pieces and summing them all up. Now you can (mathematically) prove that for any such integrable function, its integral can be approximated by Monte Carlo and the results are consistent.<p>Now what about the other direction? You can theoretically run Monte Carlo approx on wildly zigzag functions that does not make any geometry sense (i.e. not Riemann integrable), if the "probability" in the space is well-defined. The idea that uses probability, instead of geometry, turns out to give a broader class of integrable objects.<p>One interesting observation is that these ideas are intuitive and meaningful if put informally. But when you formally look into these ideas (integration/measure theory) it suddenly collapses into lines of terse mathematical constructs.