Reddit mentions of Measure, Integral and Probability

From a Micro perspective: I think the main import of an Analysis course is getting comfortable with epsilons. In particular, one should know the basics of suprema, sequence convergence (+ Cauchy), and of continuous (IVT) and differentiable functions (Rolle, MVT). You say you'll take topology, and depending on the amount of metric spaces treated there, that will actually include most of the canon on sequences and continuous functions. In differentiability, I think one can do without proofs.

I haven't seen much integration going on, so knowing Riemann or Lebesgue integration theory wouldn't be useful. I do see measure theory coming up from time to time, which is either part of advanced probability or Lebesgue Integration courses. I like the corresponding chapter in this book.

Measure, Integral and Probability Pretty good book. As an undergrad I used that for the third in a sequence for analysis. The first was real in one variable with the first 9 chapters of Bartle, and the second was multi-dim with Spivak's little white book.

Introduction to Probability and Mathematical Statistics This is another senior level math book, but can be used as a first year intro grad level book. It seems people either love it or hate it. I really enjoyed it.

A first Course in Probability This is the standard book. Personally, I hate it. But, you can't love all the books!

Hope that helps.

Awesome! As mentioned, Rudin, Folland, and Royden are the gold standards of measure theory, at least from what I have heard from professors and the internet. I'm sure other people have found other good ones! Another few I somewhat enjoy are Capinski and Kopp and Dudley, as those are more based on developing probability theory. Two of my professors also suggested Billingsley, though I have not really had a good chance to look at it yet. They suggested that one to me after I specifically told them I want to learn measure theory for its own right as well as onto developing probability theory. What is your background in terms of analysis/topology? Also, I am teaching myself basic measure theory (measures, integration, L^p spaces), then I think that should be enough to look into advanced probability. Feel free to PM me if you need some help finding some of these books! I prefer approaching this from the pure math side, so mathematical statistics gets a bit too dense for me, but either way, I would look at probability then try to apply it to statistics, especially at a graduate level. But who am I to be doling out advice?!

*Edit: supplied a bit more context.

Roughly: Limits and the Riemann integral don't go well together. You can formulate some of the theorems for the Riemann integral but only under rather restrictive assumptions (high regularity of the functions involved). And then there is the completeness (every Cauchy sequence converges) of the L^p spaces, in particular the Hilbert space of QM.

But to actually calculate a Lebesgue integral you'll usually do this by calculating the Riemann integral (in case the function is Riemann integrable) or express it as a limit of Riemann integrals.

IMHO it's totally worth to learn Lebesgue integration, the fundamentals aren't that hard (this is a very gentle introduction). It's far easier to work with. And if you are interested in the mathematical foundations of QM or probability theory it's a must.