I've studied mathematics and statistics at undergraduate level and am pretty happy with the main concepts. However, I've come across measure theory several times, and I know it is a basis for probability theory, and, unsurprising, looking at a basic introduction such as this Measure Theory Tutorial (pdf), I see there are concepts such as events, sample spaces, and ways of getting from them to real numbers, that seem familiar.
So measure theory seems like an area of pure mathematics that I probably ought to study (as discussed very well here) but I have a lot of other areas I'd like to look at. For example, I'm studying and using calculus and Taylor series at a more advanced level and I've never studied real analysis properly — and I can tell! In the future I'd like to study the theory of differential equations and integral transforms, and to do that I think I will need to study complex analysis. But I don't have the same kind of "I don't know what I'm doing" feeling when I do probability and statistics as when I look at calculus, series, or integral transforms, so those seem a lot more urgent to me from a foundational perspective.
So my real question is, are there some application relating to probability and statistics that I can't tackle without measure theory, or for that matter applications in other areas? Or is it more, I'm glad those measure theory guys have got the foundations worked out, I can trust they did a good job and get on with using what's built on top?
Best Answer
First, there are things that are much easier given the abstract formultion of measure theory. For example, let $X,Y$ be independent random variables and let $f:\mathbb{R}\to\mathbb{R}$ be a continuous function. Are $f\circ X$ and $f\circ Y$ independent random variables. The answer is utterly trivial in the measure theoretic formulation of probability, but very hard to express in terms of cumulative distribution functions. Similarly, convergence in distribution is really hard to work with in terms of cumulative distribution functions but easily expressed with measure theory.
Then there are things that one can consume without much understanding, but that requires measure theory to actually understand and to be comfortable with it. It may be easy to get a good intuition for sequences of coin flips, but what about continuous time stochastic processes? How irregular can sample paths be?
Then there are powerful methods that actually require measure theory. One can get a lot from a little measure theory. The Borel-Cantelli lemmas or the Kolmogorov 0-1-law are not hard to prove but hard to even state without measure theory. Yet, they are immensely useful. Some results in probability theory require very deep measure theory. The two-volume book Probability With a View Towards Statistics by Hoffman-Jorgensen contains a lot of very advanced measure theory.
All that being said, there are a lot of statisticians who live happily avoiding any measure theory. There are however no real analysts who can really do without measure theory.