"Real and Functional Analysis" covers a lot of material in a single volume and doesn't go into a great deal of depth on each topic. In that sense, if you just want the basics on the Lebesgue integral or spectral theory, it's a pretty direct route. I liked that the Lebesgue integral is done for functions with values in a Banach space, though I don't know how much I would have liked it if I hadn't first seen the usual presentation in Rudin's "Real and Complex Analysis". There are some fun exercises in the book, with some of them turning out to be surprisingly hard, at least for me.

I've never read "Undergraduate Analysis", but glancing at the book, it seems like Part I would be good for bringing someone up to speed even if they've never had a very rigorous course in calculus. After that, judging from the table of contents, the level seems similar to Rudin's "Principles of Mathematical Analysis" or Apostol's "Mathematical Analysis", but with a "soft" approach that's not typical for most undergraduate analysis books in English. That is, the emphasis seems to be on using general principles and topological or algebraic thinking to solve problems rather than proving many "hard" theorems in the body of the text. Again, this is based on a very cursory look at the book, and I could be wrong.