Analysis concepts relevant to probability theory

Question

I am taking a course based on Durett. I was wondering if anyone could point me to the analysis skills or topics that are most relevant to Probability Theory. It has been some time since I've taken real analysis, so was just hoping to obtain some direction.

Possibilities seem to be delta epsilon proofs, sequences (Cauchy), countability, limits, derivatives, integrals (Riemann).

Update: The answers were very good and very helpful. After having taken the course, I would like to just add some of my thoughts for any future students.

• Random variables are functions and things like expectation are just integrals on these functions. Further, things like CDFs and measures are just functions. Hence all concepts regarding continuity and behavior of functions were very relevant. In particular, probability theory ended up focusing a lot on behavior of these functions at their limits, or on behavior of sequences of these functions at their limits. Hence, analysis is central to probability theory, and one could practically learn probability theory without any background in statistics and only strong background in real analysis (although statistics experience definitely "grounds" the experience of learning probability). Proofs concerning the behavior of sequences of functions in their limits were central to most of the famous topics (eg, law of large numbers). Hence, a strong real analysis background is integral to probability theory -- I would recommend studying it beyond an introductory course (although complex analysis did not come up too often) before taking probability theory. Here is a nice video linking topics in analysis to Lebesgue integrals: https://www.youtube.com/watch?v=axEyHyIP5KM, following Durett, but with more examples and explanation. This is kind of a key topic in probability theory and can give a sense of how analysis feeds into it. He has a few others on this topic that show some very relevant concepts to probability theory. Besides real analysis, strong technical skills in calculus are important just to be able to understand examples and solve problems.
• Measure theory, as Ben pointed out in the answer below, is also key and sometimes not covered in analysis courses (although I believe Rudin covers it). I wish I had asked this question sooner and found a more fundamental measure theory textbook earlier (I got behind). My course used Durett, as mentioned above. A textbook that covers topics in measure theory more comprehensively is Resnik's A Probability Path, which I would highly recommend. It provides many of the nuts and bolts of measure-theoretic concepts that Durett assumes mastered. I guess analysis background should give some familiarity with sets, but a good prerequisite (if it's offered) would be a course in basic set theory. For example, it's important to be able to easily discern the limits of sequences of sets, to be very comfortable with things like containment, Demorgan's laws, set equality etc. We covered set theory quickly in my analysis course but focused more elsewhere than would have been helpful for this course. Resnik gives a nice introduction to this; Durett does not. Durett however seemed to incorporate less new notation than Resnik, and perhaps for that reason (and because Resnik provides more details during derivations), Resnik is quite long.
• Topology would have been useful, as mentioned below, although our professor only made reference to it during the course because it was not a required prereq. There are however many deep connections between topology and probability theory, so I believe knowing more about topology would have made the course easier
• Finally, in summary, if I had only a little bit of time and needed to review analysis (and I had Abbott's book), I would focus on having good intuition for the real numbers (basic concepts about density, rationals, irrationals), limits of sequences, subsequences, sequences of functions, the monotone convergence theorem, open and closed sets, limits, continuity, uniform convergence, Riemann integration, and Lebesgue integration. There are many curveballs that can be thrown into the lectures, but having some understanding of these core concepts can help one stay afloat.

Answer 1

Probability theory is concerned with the analysis of probability measures, which fall within the field of measure theory. If you are studying probability theory at graduate level then you will probably need to learn some of the basics of measure theory and learn real analysis and integration as an offshoot of this. This field generally involves some more abstract definitions of integration than are used in the standard Riemann integration method, and these take a while to learn.

As you are no doubt aware, the first chapter of Durrett (2019; updated) introduces probability theory within its proper measure-theoretic context, and so it makes reference to various ideas from this field (e.g., sigma-fields, Lebesgue measure, etc.). If you have not already learned some measure theory then this book is probably going to be quite a difficult read, since it moves rapidly through this material. If you find this troublesome, I recommend looking for some introductory books on measure theory, or measure and integration. Books like that will introduce you to the framework of measure theory and measure-theoretic notions of integration, and this will give you a good grounding for real analysis and probability theory.

Answer 2

You will need more than Riemannian integration if you really want to get into the theory - measure theoretic integration is the overall framework used for handling arbitrary random variables (continuous, discrete, or other) in a unified manner. That leads to some generalizations of the other concepts you mentioned, such as the Radon-Nikodym derivative. Some familiarity with Topology might be useful in the sense of prior experience with some of the more basic ideas and techniques.