How to learn measure theory and probability theory for engineering by myself?

Question

I am an engineering student and have taken courses in probability and statistics, but none of them have offered formal proofs involving constructions such as probability spaces with explicit reference to sigma algebras.

Now that I have an understanding of the applications of these techniques, I would be interested in learning more about the proofs behind them and the necessary level of rigour required to understand them (for example, I know that measure theory is an essential prerequisite to a rigorous understanding of probability theory).

Question: How might I formalise my knowledge of measure theory and probability theory on my own?

I am specifically looking for resources that I can use on my own (so taking certain university courses is not within the scope of the question). I would be interested in a range of resources - not exclusively books. For example, videos, websites, interactive tools, etc. would also be of interest.

Motivation: As an engineering student, this would be useful to me in particular as I have already used probabilistic results to predict the probability of success and failure of particular designs. Whilst my studies have led me to only look at ideal conditions (where all assumptions hold), this is insufficient for me to properly apply these concepts in a practical sense. Therefore, I am looking to make my understand more rigorous so that I can adapt and modify the processes that I am already familiar with in a way that is still mathematically coherent.

Progress: My progress so far is that I am familiar with many of the fundamental concepts in probability, but do not know the proofs and how to make my understanding of these results “more rigorous”. As a rough guide, this would make me familiar with first or second year undergraduate level probability without the foundations of measure theory.

Thank you in advance for any contributions.

Answer 1

Read the Bartle’s book about Measure and the Lebesgue Integral, lecture recommended.

You do need some prerequisites including some set theory, topology (the first half of Munkres’ book), and integral calculus.

Answer 2

A good place to start would be to read "Measure Theory" by Donald Cohn. This is a good book that walks you through all the necessary definitions with examples. This isn't a reference book, but is suitable as an introduction to measure theory.

An optional next step would be to pick up a copy of "Probability and Measure" by James Norris. This is a good reference book, but is quite dense and isn't really suitable to learn measure theory. Once you learn the tools and essentials in the book by Cohn, this book will give you some more interesting results and properties (and proofs), but is a bit more difficult to follow.

If you aren't a fan of books, then something that could supplement your learning (although I strongly recommend using the books too) is this good YouTube playlist by The-Bright-Side-Of-Mathematics. There are $23$ videos that provide some good additional assistance in explaining difficult concepts and proofs.

After this, you will have strong enough fundamentals in measure theory in order to tackle a rigorous course in probability theory. As mentioned in the comments, the standard text for a study of probability theory is the book "Probability and Martingales" by David Williams. Like the book by Cohn, this is not a reference book, but is a good book for self-study.

A difficult, but more in depth, treatment of probability theory (and even some stochastic calculus) is given in "Probability Theory: A Comprehensive Course" by Achim Klenke. I would recommend working through the book by Williams first as there are more examples and there is more time taken to explain the concepts than the book by Klenke - which is designed for a postgraduate treatment of probability theory.