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Comment by benrbray | Hacker News Reader

For anyone looking properly (i.e. with proofs) learn measure theory, I highly recommend "Introduction to Measure Theory" [1] by Terrence Tao (has an excellent collection of exercises -- they're crucial to the reading!)

I think it's OK to skim the first few chapters about constructing Lebesgue measure from Jordan measure, as it makes the whole topic appear more difficult than it needs to be. From Chapter 4 (abstract measure spaces) onwards, the book is full of great insight.

A good one on probability & stochastic processes is "Measure Theory and Probability Theory" by Athreya & Lahiri.

[1] https://terrytao.files.wordpress.com/2011/01/measure-book1.p... [2] https://www.springer.com/gp/book/9780387329031

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I’d just make a slight correction in case anyone was confused. The book actually only has two chapters, one on the Lebesgue Measure and one on Abstract Measures. When the person above me said skim the first few chapters they meant sections of the first chapter. I’ll admit it’s a somewhat confusing way to lay out the book.

I also can’t recommend this text enough for anyone looking to learn measure theory. Tao is not only one of the brightest research mathematicians but also is extremely good at communicating and framing concepts.

How useful do you think measure theory is from a practical perspective? I've seemed to get by without more or less during my career (which is pretty stats heavy).

Learning measure theory has eliminated a lot of blind spots for me when it comes to stats / probability / ML. I find that I get "stuck" much less frequently when reading papers, and I'm able to better in the gaps when authors are too sparse with the mathematical details. Before, I would always wonder, "how on earth did they come up with this idea?!", but the more math I learn, the more I'm able to recognize that certain ideas are inspired by XYZ branch of math.

It's also been useful for understanding the "novelty" papers like "Neural Differential Equations" and so on.

I was lucky enough to take a good measure theory class in college, where it was fine to spend an entire day working through a proof from a textbook. Now that I'm working, though, it would be harder to justify that time commitment. So it really depends on your priorities.

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