The mean deviation of a function on independent trials

Question

I have a function f(x).

I do M runs and M tends to infinity. For each run, I generate two random samples from the domain of f and calculate the square of the deviation.

The image might help clarify.

Equation

What I have found is that the integral/sum over the uncorrelated, independent deviations adds up to 0. An explanation of why this is would be really helpful.

Thanks