I haven't read the rest of the paper linked by blog post (and I know next to nothing about machine learning itself anyway), but the page linked clearly suggests the motivation behind Lebesgue measure: the theorem that author wants to prove says that some sequence of random random variables ve defines is almost surely convergent, and the limit is almost surely equal to some other random variable ("almost surely" is another mathematical term with a well-defined meaning that can be very confusing, but this one is actually easy enough to make Wikipedia explanation understandable). There are some strong and useful theorem for the convergence of the limits of Lebesgue integrals, and there are also theorems that sometimes guarantee that whenever intervals converge, functions will converge as well. The author probably wants to use these theorems.

Now, could it all be done without mentioning Lebesgue integral and Lebesgue measure? In most cases, yes, in this case almost surely. The question is, would it actually be easier and simpler? The author would probably need to reproduce the ideas needed for the proofs of convergence theorems for Lebesgue integrals, which wouldn't be really important and relevant. It's much easier to put your problem in setting in which it has already been solved.

Anyway, as I mentioned above, every mathematician is intimately familiar with Lebesgue measure and integral, so putting the problem in some different, "simpler" setting than this would probably make it more difficult to understand.