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?