# Analysis concepts relevant to probability theory

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.
• And also sigma-algebra
– Sycorax
Aug 30, 2018 at 22:58
• Probability measures are not "just functions on the real numbers," I'm afraid: they are functions on algebras of subsets of numbers. This misconception mars several parts of your edit, which otherwise looks to be well-considered and helpful. (+1)
– whuber
Jan 15, 2019 at 13:29
• @whuber, thanks and appreciate the feedback - took some sentences out which hopefully fixed it, will leave descriptions of measures/etc for another question Jan 15, 2019 at 18:59