# Preparing for background to understand measure theoretic introduction to probability

I believe this is useful for many students/researchers who come from related fields like computer science, and have not had the chance to take pure math courses in analysis.

My situation is as follows. I am a beginning graduate student in computer science. I want to take the measure theoretic probability course that starts in the beginning of September (and follows Rick Durrett's "Probability: Theory and Examples"). I really want to do it. So that leaves me with close to 3 months. And my current background in mathematics is limited to linear algebra/calculus/intuitive probability. Although I haven't done pure math, I have taken complexity theory course in computer science, and am comfortable with writing rigorous proofs and dealing with abstract concepts.

Could someone advise me the steps to reach the stage where I am able to understand the course? I have been going through Spivak's calculus to revise the feel of rigorous calculus. Should I go through "baby Rudin" next? Which chapters are enough? Is that all, or do I need to know more? I need to know all this in advance so that I can plan for the next 3 months. Is there an online set of lectures/notes that could be helpful?

Maybe it is useful to add that I am going to be simultaneously attending 2 semester standard real analysis sequence that begins with baby Rudin. But I want to be prepared for the probability course in advance.

Thank you.

• You'll need to understand introductory real analysis, mostly metric spaces, continuity, completeness, convergence of sequences, $\limsup$ and $\liminf$, some basics of point-set topology, and be comfortable with reading and writing proofs. – Math1000 Jun 6 '18 at 20:23
• Thank you @Math1000. So you think first 7-8 chapters of baby Rudin will suffice? Do I need to learn Lebesgue measure or that can be picked up as some measure theory is reviewed in first week or so of the probability theory course? – student Jun 6 '18 at 23:28
• Rudin is a classic text but I wouldn't recommend it if you are completely new to analysis. It is a bit terse and the proofs, while elegant, can be difficult to follow. I like the text Introduction to Analysis by Rosenlicht. It is also rigorous but lends much more intuition to the theorems etc. – Math1000 Jun 7 '18 at 5:37

## 1 Answer

When I was a student, I followed a course on real analysis that went through the first eight chapters of Principles of Mathematical Analysis (aka "baby Rudin") – and after that came a course on measure theory using Real and Complex Analysis, also by Walter Rudin. This was the intended progression, and it went well (at least for me).

The most important prerequisite, however, for a measure-theory-based course on probability theory, is that of mathematical maturity. You must be able to read and understand definitions and theorems and to be able to follow the structure of proofs. This is not a trivial undertaking, and many computer science students find this to be quite the challenge.

• Thank you @Hans. Have you taken a computability theory/complexity theory/algorithms-analysis course? I ask this so that you can give me insight into whether I have sufficient maturity or not. Like what is a "test" that can tell me whether I have the required maturity or not? – student Jun 6 '18 at 21:09
• Indeed I have. I am in a computer science department. If one can pass an advanced course in computational complexity or computability theory, I see no problems. – Hans Hüttel Jun 6 '18 at 21:36
• Thank you! I like to think it was of reasonable difficulty, like beginning grad or advanced undergrad: we covered most of the Sipser's third edition, including P/NP/NP-completeness, PSPACE, L/NL, oracles, BPP/ZPP/RP, NC, and interactive proof systems, not to mention automata theory, TMs, decidability/r.e.. – student Jun 7 '18 at 1:58