This is a text for a one-year course in real analysis for graduate students (or for very good undergraduates). Reviews that give tables of contents can justifiably be called boring, but someone looking for a text for a course whose content is not the same at all times and in all places needs to know what it contains, so look at the table of contents that's provided by MAA Reviews.
I should add to the listing of contents that the Lebesgue integral is done using Riesz's approach, that is, using the order integral, then measure, rather than measure, then integral.
The treatment is clear and thorough. Everything is here. The author includes exercises in the text that readers are expected to do, or at least think about, as they go along. There are extensive problem lists at the end of chapters, which is a way of guaranteeing that nothing will be left out. For example, after the last chapter there are one hundred and fifteen problems, including Egorov's Theorem, the Hölder and Minkowski inequalities, and a good deal of theoretical probability. This is a very rich book. Serious students would benefit greatly from it.
Traditional reviews start by giving the table of contents and end by noting two or three misprints that the alert reviewer has noticed. Well, I didn't see any misprints, nor any grammatical errors. The book is a pleasure to handle, physically and intellectually.
Woody Dudley, whose training in real analysis dates back forty-seven years, can no longer do one hundred and fifteen problems but sees no reason why those younger and more vigorous shouldn't be required to.
Sequences and Series of Real Numbers
Limits of Functions
Topology of R and Continuity
The Riemann Integral
Sequences and Series of Functions
Normed and Function Spaces
The Lebesgue Integral
General Measure and Probability
Appendix A: Construction of Real Numbers