Preface In the fall of 2010, I taught an introductory one-quarter course on graduate real analysis, focusing in particular on the basics of measure and integration theory, both in Euclidean spaces and in abstract measure spaces. This text is based on my lecture notes of that course, which are also available online on my blog, together with some supplementary material, such as a section on problem solving strategies in real analysis (Section 2.1) which evolved from discussions with my students. This text is intended to form a prequel to my graduate text [Ta2010] (henceforth referred to as An epsilon of room, Vol. I ), which is an introduc- tion to the analysis of Hilbert and Banach spaces (such as Lp and Sobolev spaces), point-set topology, and related topics such as Fourier analysis and the theory of distributions together, they serve as a text for a complete first-year graduate course in real analysis. The approach to measure theory here is inspired by the text [StSk2005], which was used as a secondary text in my course. In particular, the first half of the course is devoted almost exclusively to measure theory on Euclidean spaces Rd (starting with the more elementary Jordan-Riemann-Darboux theory, and only then moving on to the more sophisticated Lebesgue theory), deferring the abstract aspects of measure theory to the second half of the course. I found that this approach strengthened the student’s intuition in the early stages of the course, and helped provide motivation for more abstract constructions, such as Carath´ eodory’s general construction of a measure from an outer measure. Most of the material here is self-contained, assuming only an undergrad- uate knowledge in real analysis (and in particular, on the Heine-Borel the- orem, which we will use as the foundation for our construction of Lebesgue ix
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