Cambridge University Press Probability with Martingales

I had somehow studied the topic(s) during university and read that book a few years later, working as an engineer. I am not a professional mathematician. This book was quite a thrill, it really tied the pieces together and I believe I have a much deeper understanding after reading it. The lively aspect, I would say, manifests itself in that the examples chosen really drum up interest and the conciseness allows one to have an overview. It's fairly accessible, proofs included - and even if not, any hole can easily be bridged with the help of stack/mathoverflow nowadays. Some exercises are hard though, but then again it's alright, it's not the book on continuous processes from the same author!

This book is extraordinary, beautiful and clear exposition, thoughtfully chosen exercies which increase the understand of the topic instead of just moving the reader through the list on autopilot (which many textbooks unfortunaltely do). My research is mainly in PDEs and Harmonic Analysis, and this book was one of my first (proper) introduction to martingales. Based on my own experience, this is an excellent introduction for anyone starting on probability and martingale theory. I simply can't recommend this book enough.

No disappointment from this book even if an appendix with the solutions of the exercises would have been a plus Very deep in the probability, crystal clear, good tips

測度論から確率論、マルチンゲールの話まで書かれています。 測度論は確率論へのつながりを意識しながら、必要かつ十分に説明されています。 （古典）確率論に関する部分（大数の法則や中心極限定理など）に関しては割とあっさり記述してあるように思いました。 マルチンゲールに関する部分は非常によく書けていると思います。 条件付き期待値の説明から始まり、マルチンゲールの基本的な性質、一様可積分なマルチンゲールの性質と続き、離散マルチンゲールに関して必要なことは一通り網羅されています。 この本で離散マルチンゲールを勉強した後にKaratzas-Shreveの本などに自然につながっていきます。 測度論に関する詳しい証明などは最後に付録としてまとめられていますが、ここもきちんと読んだ方が良いでしょう。 また、この本は非常にウイットに富んでおり、思わず笑ってしまうような記述が各所にみられます。非常に楽しく勉強できる本だと思います。 なお、この本には日本語訳も出ています。

The clarity of exposition and overall rigor by themselves would already make this the perfect starting point to delve into rigorous probability theory, but the way the text is self-contained is what really is astonishing. It only assumes some ideas from basic topology, calculus and fundamental real analysis (most of the stuff is included in what everyone would cover in any engineering bachelor) + a general familiarity with basic probability theory (i.e. a non-measure theoretic exposition), but the rest is really not taken for granted (and if it is, it is mostly proved and expanded in appendices), with the first 8 chapters dedicated to a strong interlacing between real analysis and probability theory, alternating between the two things and always letting the reader understand the way the former helps formalize and expand the latter. Clearly, it is a quite technical textbook, so it requires a notable amount of effort especially in some sections; however, you never get really stuck since the writing and the flow of the proofs is really accessible. D. Williams really wrote a masterpiece.