Congrats on some insight into Rudin! He got less terse over time. His Real and Complex Analysis is terrific -- hardly terse at all. The only question I had w

Congrats on some insight into Rudin! He got less terse over time. His Real and Complex Analysis is terrific -- hardly terse at all. The only question I had was, why regular Borel measures? Sure, if I went back and studied again where he used the hypotheses I'd see it. But in Rudin's favor, e.g., for his Principles, it's all there, and crystal clear, very precise, and perfectly correct. Yes, have to draw own figures, at least between ears, maybe on paper. Students might be told that one of the purposes of the early material in Rudin's Principles is to build a really solid foundation for discussing continuity and uniform continuity. So, for that, he wants compactness. He uses that work to show that, with continuity on a compact set and, thus, uniform continuity, the Riemann (Riemann-Stieltjes) integral exists. Then near the end of the book he shows that the uniform limit of continuous functions is continuous. But, also he goes ahead and shows that a function has a Riemann integral if and only if it is continuous everywhere except on a set of Lebesgue measure 0, without really saying much about Lebesgue measure. I don't have much from Apostol. I'm sure he could write a book that competes with Rudin's Principles and might be easier for ugrads to read. It happens that two weeks or so I got out Rudin's Principles, the third edition, and looked up something and was struck at just how beautifully, elegantly, sparsely done the work actually is. On elegance, Neveu is my favorite.