should I learn measure theory before learning probability?

Question

I am currently looking to learn about probability and statistics since I am interested in actuarial science. I have some knowledge on real analysis(rudins book except the last 2 chapters) and linear algebra(axlers linear algebra done right). I have very little prior knowledge about prob/stat.

When researching prob/stat books to order I encountered the distinction between books that use measure theory and those that don't.

Anyway I am not really sure where to start and was wondering if someone could kindly recommend some books and which order to read them in.

Answer 1

In fact, it's the inverse. Try some introductory probability books (e.g. Kai Lai Chung's introductory probability book), before beginning real analysis. In that way, you know the motivation for studying abstract integration. If you want an introductory book with more discussions on measure theory, try David Pollard's A User's Guide to Measure Theoretic Probability.

Answer 2

Quoting Rick Durrett from his book Probability: Theory and Examples, "Probability theory has a right and a left hand. On the left is the rigorous foundational work using the tools of measure theory. The right hand 'thinks probabilistically', reduces problems to gambling situations, coin-tossing, and motions of a physical particle."

A lot of probabilistic principles can be learned from finite or countable sample spaces, for which essentially no measure theory is required. Ross's a First Course in Probability can be profitably read without any measure theory. Once you start learning about things like Brownian motion, you'll find that measure theory becomes unavoidable to define the concept precisely. But even there, thinking about Brownian motion as just a discrete random walk with the mesh size approaching 0 can get you quite far.

Answer 3

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.

Answer 4

The new book on measure theory that I am writing may be useful to you. It's title is Measure, Integration & Real Analysis. The first eight chapters are currently freely available on the book's website: http://measure.axler.net/. More chapters will be available on the website as they are completed.