Real Analysis for Graduate Students, by Richard Bass (download for free online from http://bass.math.uconn.edu/real.html)
No prerequisites for Math Ph.D. students; permission of instructor otherwise.
Basic real variable function theory, measure and integration theory prerequisite to pure and applied analysis.
Topics: Riemann and Lebesgue-Stieltjes integration; measure spaces, measurable functions and measure; Lebesgue measure and integration; convergence theorems for integrals; Lusin and Egorov theorems; product measures and Fubini-Tonelli theorem; signed measures, Radon-Nikodym theorem, and Lebesgue's differentiation theorem.
Note: A Problem Session for 501 will be held on Wednesdays, 10:00am-11:40am in room 425 for the Fall semester. There is no need to register for the Problem Sessions.