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A professor told me it would be unwise to study an $\mathbb{R}^{n}$ analysis book that uses Riemann integral. However, I do not know any book that doesn't. Is there something that covers material similar to Spivak's Calculus on Manifolds, but using Lebesgue integrals instead? I don't think these topics are generally covered in a measure theory book.

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Andrew Browder's Mathematical Analysis does multivariable integral calculus using the Lebesgue integral. It is often overlooked because of classics like Rudin's Principles of Mathematical Analysis. It treats point-set topology, differential forms and Stokes's theorem too, totally awesome!

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  • $\begingroup$ This looks nice. Maybe a bit repetitive since I'm already familiar with the first half of the book (up to and including topology), but if nobody suggests something better I might get it. $\endgroup$ – Pedro Aug 16 '13 at 18:25

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