A lot of measure theory-oriented books I've seen seem to presuppose plenty of familiarity with topological/set theoretic concepts and notation. For instance, when using Folland's "Real Analysis" in grad school for learning Lebesgue integration, I was totally unprepared for the motivational discussions about uncountable and unmeasurable sets, even though I had some prior familiarity with infinite sets and the basic pathologies that can arise in them (e.g., Cantor set). That made getting through even the first couple chapters really difficult because I felt like I was groping around in the dark and just carrying out formal manipulations without a clear sense of the obstacles that these advanced tools were being developed to overcome. A brief look through the intro of Pollard's book (recommended above) suggests to me the same issues.
As such, I'd recommend working through an undergraduate-level Topology text before approaching anything with measure theory. I've been doing that with S. Morris's "Topology without Tears" (free online!), and it's really helped me flesh out how much variety there is in general spaces before we even get to the notion of a metric. I feel like I'm almost ready to revisit Folland--just after I finish Morris's chapters on metric spaces and compactness. This also dovetails nicely with Axler's "Linear Algebra Done Right", since it gives another side of the story motivating the development of different kinds of norms.
[Edit: Actually, I'm just about done with Morris's chapter on Metric Spaces, and I must say that, compared to the rest of the book so far, I'm not terribly impressed. Admittedly, he does say that MS theory is its own field separate from topology, so that make the lack of clarity a little forgivable. Still, it's annoying to have the hypotheses and specific definitions in theorems/corollaries and problems not clearly stated; maybe it's just me, but this seems to be a real difficulty in section 6.5 on the Baire Category Theorem. Anyway, I think I'm just going to skip the rest of this chapter and move on with the book.]
Also, since you're looking at statistical issues, I'd also recommend reading through the first couple of chapters of E.T. Jaynes's "Probability Theory: The Logic of Science", since he gives a very accessible description of a lot of fundamental issues in probability/statistics that are often hand-waved away in introductory treatments.