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.
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.
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.
If you want all subsets of sets with 0 Borel measure to also have 0 measure, then this leads to the notion of the lebesgue measure, and it can be used to define the lebesgue integral.
a) But for probability you nees the measure of the whole space to be 1, which forces points “far away” to have very small “weight”. Thus, there is no uniform distribution in the whole R.
b) and then your mind blows up when you realize that discrete probability is the very same thing, and that an integral in a finite set is just summation.
My only reservation was that it updated at a glacial pace for topics I really wanted. I was waiting for volume rendering 4 years ago, and it looks like they started making articles last month. But I can't really complain about updates to completely free resources.
Have a nice weekend, regards...
