# should I learn measure theory before learning probability?

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.

• The last two chapters of Rudin do a great job motivating measure theory. – Don Thousand Dec 4 '18 at 17:20
• Learn them at the same time. – Shalop Dec 4 '18 at 17:35
• Can you help the mathematician s by explaining what one studies in actuarial science. Eg if you only work with discrete distributions measure theory is irrelevant – seanv507 Dec 4 '18 at 23:31
• Another approach is to go through a book that introduces both at the same. Williams' Probability with Martingales is a fine textbook. – twnly Dec 5 '18 at 0:23

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.

• +1 for suggesting the reverse order. (I can't speak to the individual texts you recommend.) – Ethan Bolker Dec 4 '18 at 17:12

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.

• Commenting because my class is using this book for the sequence in Real Analysis and Functional Analysis. Great book. I highly recommend it to anybody curious and looking for a good grad-level analysis book. Thanks for writing it! – Matthew Graham Mar 19 at 16:49
• I love your textbook ! I'm currently studying it and planning to buy hard copy as well once it's published – ElleryL Apr 18 at 3:18

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.

• Ross' book (in your answer) assumes that all subsets of the sample space Ω is measurable to avoid measure theory. That's fine for undergraduate statistics majors, but we all know the inconvenience of discussing mathematical ideas in imprecise mathematical language. IMHO, Chung/Pollard/other introductory probability books that adopt Kolmogorov's axiomatic definition of probability are much better choice. – GNUSupporter 8964民主女神 地下教會 Dec 4 '18 at 19:57
• OP mentions interest in actuarial science. I agree that any serious probabilist or theoretical statistician will eventually need a solid grounding in the logical foundations. But honestly, there are many bright people in industry and even applied statisticians in academia who solve sophisticated problems involving probabilistic reasoning and wouldn't be able to state Kolmogorov's definition of a probability space. A famous statistician once said, “I wouldn't want to fly in a plane whose design depended on whether a function was Riemann or Lebesgue integrable." – zoidberg Dec 4 '18 at 20:16

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.

• Topology without Tears can be a soft introduction. Personally, I can't find another introductory topology book better than Munkres's Topology. – GNUSupporter 8964民主女神 地下教會 Dec 4 '18 at 19:44