I find I can not really understand the Monte Carlo integration, even I use it for many applications, like stochastic ray tracing.
Let us take circle-area-calculation for an example,
First, we think about the familiar integration method: Informally speaking, the integration is limit of infinite sum, and intuitively, every summand(or integrand) should have same dimension(unit) as the result of integration.
In the circle area calculation example, every summand(infinitesimal area patch) has the same unit as result of integration(that is, $m^2$).
But in Monte Carlo integration, I find it is integrating a function of random variable. This makes me confused. Because I think the random variable has no unit/dimension.
Given dimension-less(I am not sure is it dimension-less) integrand, why does the MC integration finally converge to a value represents "area", which has a unit of $m^2$?