I am no expert here, but as I understand it, measure theory is used as the basis for all modern probability theory for the reason that it simplifies a lot of th

I am no expert here, but as I understand it, measure theory is used as the basis for all modern probability theory for the reason that it simplifies a lot of things as you get to more advanced topics, like Markov chains on general state spaces [1]. So if you want to study Stochastic Processes you probably want to use measure theory. So thus the Lebesgue integral is not just for "pathological" sets (I remember reading somewhere that even the Lebesgue integral still breaks down on some really pathological examples). If you've studied mathematical statistics, you'll typically see proofs for expectations of discrete distributions and continuous distributions. They're typically similar, but different since you're using summation for the one, and integration for the other. When you get to random vectors, you can have weird distributions where some components are discrete and others continuous. Apparently using a measure-theoretic approach to probability, it unifies these into one general theory (again, I'm no expert, this is how I understand it). [1] https://www.stat.umn.edu/geyer/8501/measure.pdf https://www.stat.umn.edu/geyer/8501/measure.pdf