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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)?

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    $\begingroup$ Conditioning for continuous variables, stochastic processes come to mind as cases requiring measure theory for a rigorous definition. In my maths program in Paris-Dauphine, measure theory is actually part of the undergraduate (compulsory) curriculum. $\endgroup$ – Xi'an Dec 5 '19 at 9:46
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Two examples:

All of functional analysis, which I guess you will know underlies a lot of machine learning, relies on measure theory. There is no "undergraduate" probability measure that describes a distribution over function spaces, as far as I'm aware.

Statistical analysis of nonlinear dynamical systems - there is generically at least one starting condition for a dynamical system such that the infinite-time evolution of the system lies on a set of measure zero in state space. Statistical analysis of such systems can go badly wrong without rigorous measure theory.

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