This is not going to directly answer your question, but it's too long for a comment.
From Chapter 1 of "The Theory of Statistics
and Its Applications" (https://www.stat.rice.edu/~dcox/Stat581/chap1-2.pdf):
Measure theory is a rather difficult and dry subject, and many statisticians
believe it is unnecessary to learn measure theory in order to understand statistics. To counter these views, we offer the following list of benefits from studying measure theory:
(i) A good understanding of measure theory eliminates the artificial distinction between discrete and continuous random variables. Summations become an example of the abstract integral, so one need not dichotomize proofs into the discrete and continuous cases, but can cover both at once.
(ii) One can understand probability models which cannot be classified as either discrete or continuous. Such models do arise in practice, e.g. when censoring a continuous lifetime and in Generalized Random Effects Models such as the Beta-Binomial.
(iii) The measure theoretic statistics presented here provides a basis for understanding complex problems that arise in the statistical inference of stochastic processes and other areas of statistics.
(iv) Measure theory provides a unifying theme for much of statistics. As an
example, consider the notion of likelihoods, which are rather mysterious in
some ways, but at least from a formal point of view are measure theoretically
quite simple. As with many mathematical theories, if one puts in the initial
effort to understand the theory, one is rewarded with a deeper and clearer
understanding of the subject.
(v) Certain fundamental notions (such as conditional expectation) are arguably not completely understandable except from a measure theoretic point of
From the start of https://statmodeling.stat.columbia.edu/2009/05/27/the_benefits_of/
Stephen Senn quips: “A theoretical statistician knows all about measure theory but has never seen a measurement whereas the actual use of measure theory by the applied statistician is a set of measure zero.”