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Yes, I know that this question has been asked before, but let me explain why I am asking again.

I have had the experience of taking a measure theory course using the book by Donald Cohn on the subject, and the exercises were tough.

Let me explain some features of a measure theory book I would be interested in:

  • exercises of various levels of difficulty (if possible, in sorted order or marked to indicate difficulty level)
  • a solutions manual to (at least some of) these exercises
  • would like the book to explain some of the tricks used in harder analysis problems
  • would like the book to have good exposition that helps to build intuition
  • would like the book to have most of the standard material on measure theory required for graduate mathematics

I don't claim that these criteria are anything more than vague, but I hope that they give a sense of what I'm looking for. For example, I liked Aluffi's Algebra book, because it has many of these feature with out sacrificing too much rigor/material needed in a graduate textbook. If possible, it'd be great for this book to cover the needed material on measure theory required for graduate mathematics. I mention Probability Theory in the title as well, on the off chance that you happen to know of a book that does these things for Probability Theory.

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I recommend for

Measure Theory: Real analysis, Gerald Folland. An Introduction to measure theory, Terence Tao.

Probability Theory (first course): Probability theory and examples, Rick Durrett. Probability with martingales, David Williams.

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