Comparison of five books on Measure Theoretic Probability.

Question

I am reading measure theoretic probability for the first time. I wish to know how the following books compare with each other. My final goal is

1. to understand many research papers in electrical engineering which use rigorous probability (like spectral shaping of quantization noise etc)
2. to be able to understand Floquet theory which is used by circuit simulators to model phase noise of circuits. Specifically, I need to understand stochastic differential equations
3. to be able to understand the nuts and bolts of various machine learning models ie the data science part of machine learning

The five books I have in mind are the following:

1. Athanasios Papoulis and S. Unnikrishna Pillai, "Probability, Random Variables and Stochastic Processes"
2. Grimmett & Stirzaker, "Probability and Random Processes"
3. Sidney I. Resnick, "A Probability Path"
4. Patrick Billingsley, "Probability and Measure"
5. David Williams, "Probability with Martingales"

I have a hunch that the above lists it in increasing order of detail and difficulty. Nevertheless, given that my goal is as listed above, please let me know as to what each book has to offer.

Answer 1

All three of your goals will require material beyond any of the five books you mentioned. The Papoulis-Pillai book is the only one that discusses the spectral theory of continuous-time stochastic processes, though it's light on the measure theory. Of the five, Resnick and Williams cover the least material, and I like Williams a lot better -- "Probability with Martingales" is a beautiful book, no matter your intended application, as is the Grimmett and Stirzaker book. Billingsley definitely demands the most maturity.

So, my recommendation: get a good measure-theoretic foundation with Williams and with Grimmett and Stirzaker, then turn to Papoulis and Pillai for the spectral techniques (if these are the only books you're going to consider). Set Billingsley aside until you have more experience working with measure theory -- I think it's a bit of a difficult book, though a very good one, unless you know enough to know what you're looking for.

Answer 2

Regarding point 3, "to be able to understand the nuts and bolts of various machine learning models ie the data science part of machine learning" , I think that any good book on probability covers the material needed to be able to start reading research papers .

But if you look for a book that is written with data science/ML in mind , I like this two :

1.https://www.amazon.com/Probability-Analysis-Data-Guy-Lebanon/dp/1479344761 (http://theanalysisofdata.com/probability/0_2.html )

2.https://www.math.uci.edu/~rvershyn/papers/HDP-book/HDP-book.html

The first one is at the undergraduate level and the second one is more advanced (but quite easy to read) and covers many topics and applications relevant for current theoretical research in data science ,machine learning and signal processing .