I've been thinking of buying books from amazon to self-study analysis. I find a lot of positive/negative reviews about either Rudin (too difficult), or Abbott (too basic) plus Kolmogorov book: Kolmogorov book. It is for self study.
I am not a mathematics major, I am computer science student, who deals with a lot of probability in machine learning. I would like to study analysis (or rather measure theory but as I understand it is part of analysis) to apply to this particular field, and to exercise to write simple proofs, and understand articles dealing with epsilons and sets (i can do basic ones from strang linear algebra and multivariable calculus courses, and geometry at school) and I would like to understand probabilistic articles on Wiki that deal with real analysis.
So can someone familiar with both options (Rudin vs abbott + Kolmogorov) advise which one is better in this regard?
Or maybe there is some other book I don't know of – but everywhere I go it is always Rudin.
This question is different from similar ones here because field of application of real analysis is gonna be probability.
PS. Thank you for recommendations here. I decided to go with Abbott for the beginning and then continue with Folland/Rudin, where the second one I found at my university library. Maybe will take a year/1.5 all in all 🙂
Best Answer
This is probably better as a comment, but too lengthy.
Abbott is a junior level analysis text. Baby Rudin is definitely a senior level analysis text, designed for a two-semester course. Rudin is terse and makes you work for it. I felt the book was too difficult when I was going through the course; but in hindsight, it was appropriate (which means I grew from the course!).
Senior analysis has the burden of building one's mathematical maturity for graduate studies. When I took senior analysis, we were expected to put in 20 hours each week. There were several grad students from other departments (at least for the first semester), that were taking this course to improve their writing skills for publishing research papers. From this perspective, Rudin was appropriate.
Measure theory is usually taught as a graduate course. It doesn't take much work to define a measure and derive basic properties; only basic set theory and basic algebra are involved ($\sigma$-fields are basically free algebras). Once you start defining the Lebesgue integral or talking about modes of convergence, having a stronger analysis background will be helpful. Graduate level probability theory classes usually introduce the requisite measure theory upfront (from what I have heard). So if your goal is probability, I would try and find a graduate probability theory course website or text to help guide you.
I would personally suggest Rudin for the rudiments of analysis. You will definitely want to cover series and sequences, point-set topology, continuity, differentiation, and series and sequences of functions. It may be safe to skip the derivative as a linear operator and the Riemann-Stieltjes integral (though these are always good things to know, especially thinking about the derivative as a linear operator, if you are doing machine learning and stochastic optimization). I would avoid Baby Rudin for measure, as I have not heard good things about his exposition.
Trench is another alternative to Rudin which I have heard recommended. I am not familiar with this text personally.
If you find Rudin too terse, you can always pick up Abbott.
Bartle's The Elements of Integration and Lebesgue Measure is a friendly text for measure theory, though it does assume some maturity. Again, measure is really a graduate level topic.
Also, if you are still a student, it might be worth sitting in on an analysis class at your institution. Having someone with intuition and classmates with whom to converse will definitely be helpful for learning the material.