Could you give me advice on self-study for topology, measure theory and functional analysis?

Question

I would really love to understand measure theory, as I want to become better at probability and statistics.

I heard that in order to delve into measure theory I need to first study functional analysis and topology. So I would love to get advice on measure theory, functional analysis and topology.

Mainly, I want these questions answered (naturally any other relevant information is more than welcome):

1. Which are good study books? Which books do you recommend for each of these three courses. Please keep in mind they are for self-study, hence I would really love a book that has solutions available (be it an official solution manual, good online community or website with solutions). Also, I know from experience that there are very good books on statistics but they are not great for self-study. But more for additional reading next to the lecture or for later in your career to look stuff up. So I would want to avoid such books for self-study.

2. Which order do you advice me to study these 3 courses in?

3. Do I need to study any other topics before delving into these three? I heard, that calculus on manifolds maybe is necessary as well? I do have a decent background in linear algebra and calculus (multiple university courses).

Answer 1

The only prerequisite for measure theory (in the context of probability theory) is a solid understanding of real analysis (say, where one treats general metric spaces), and definitely not functional analysis or general topology. I say this as someone teaching a measure theory course before the students even have had a topology class. It is true however that there are lots of connections with both other areas, but that's a different question.

My favorite references:

• Very good for a beginner, and free: the recent book "Measure, Integration and Real analysis" by Axler. This book discusses some basic functional analysis at some point later in the book, but everything is done from scratch so that shouldn't be a problem. It is also not necessary to read this part to understand the measure theory.

• "Measure theory" by Cohn. A more advanced measure book, treating more advanced topics. Ideally you use this after Axler's book.

• "Real analysis" by Folland. Also more advanced and a second good book.

The focus in the above books is not on the probability theory. If you really want to see measure theory from a probabilistic point of view, I can recommend:

• "Measure and probability" by Billingsley.

Answer 2

I wouldn't say you need Functional Analysis to study Measure Theory. You do need a bit of General Topology though. So, in terms of study order, I'd advise going through the basics of Topology first, then study Measure Theory and, finally, have a look at Functional Analysis.

For Topology, one of the best references I can give in Munkres' book "Topology". I'd say read Chapters 2 and 3, and visit Chapter 1 when needed. To start looking into Measure theory this is more than needed. Also, you can find a lot of solutions on-line...

In terms of Measure Theory, I agree that Folland's book "Real Analysis" is a good reference, but it might be a bit tough depending on your mathematical maturity. You could also try "The Elements of Integration and Lebesgue Measure" by Bartle.

Finally, for Functional analysis a standard reference is Kreyszig's book "Introductory Functional Analysis with Applications".

I would like to add a reference for Probability Theory as well. Have a look at Alan Gut's book "Probability: A Graduate Course". This one presupposes Measure theoretic knowledge, hence the word "graduate".

Of course all of these recommendations depend immensely on the knowledge you already have. If you find these subjects too hard to follow, then it's probability because you don't have enough mathematical training yet. Which is fine! Simply start with simpler subjects like Linear Algebra and Calculus. Mathematics takes time, especially when you start to get into these kinds of subjects.