$\mathbb{R}^{n}$ analysis book that uses Lebesgue integrals 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.

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!