Where is measure theoretic probability theory actually applied?
I've done quite a bit of graduate work in machine learning, Bayesian machine learning, information theory, and statistics (both Bayesian and frequentist), and I've never found that I actually had to apply measure theory.
I'd been exposed to it when learning about stochastic processes, but it wasn't clear why we actually needed it.
So, is there any field of study where you can't proceed unless you use measure theoretic probability theory (as opposed to typical undergraduate probability)?