Measure Theory by Axler

After reading my first Axler’s book, Linear Algebra Done Right, I felt his writing style is my way of learning things and was motivated to read all his books. One that was very relevant for my Ph.D research topic was obviously measure theory. Some background, I read parts of Billingsley’s "Probability and Measure", Feller’s “Probability Theory” (recommended by Nassim Taleb), Ramble through Probability, Ross’s “Probability and Statistics”, and a few others. Nothing stuck like Axler’s way of writing. Yes, I agree Billingsley is thorough, rigorous, and arguably tougher questions, yet its dryness would be suited for a classroom not a self-taught environment.

I also believe this is a great book for revisiting classic topics like spaces. The book’s superpower comes in the form of foreshadowing. I think this idea of laying the path in steps blind is not the way to learn any topic in math. The author is great to notify the reader once they come upon a lemma that seems simple and unassuming (lets say Zorn’s Lemma) will be the reason behind very important results. (who knew having a maximal element would give you the existence of a basis, and every Hilbert spaces have orthonormal basis).

This is a book where I cannot really suggest to read snippets from. The way its laid out requires previous references and I would not recommend swifting through or jumping sections without having a solid understanding of previous sections. An absolute must to read the topic of measure theory for many practitioners and students, and this book is a delight for that mission.