Bass R. Real analysis for graduate students

University of Connecticut, 2020. — 474 p.Preface to Version 4.1The biggest change is that I added a Chapter 27, which gives solutions and hints to selected exercises from Chapters 2–19. The exercises I selected were based on: is this an important result, is this hard, is this a representative problem ? I changed the proof that the trigonometric polynomials were dense to avoid the use of the Stone-Weierstrass theorem. The proof is now more elementary. I also reworked Chapter 14 to make it more streamlined.Preliminaries.
Families of sets.
Measures.
Construction of measures.
Measurable functions.
The Lebesgue integral.
Limit theorems.
Properties of Lebesgue integrals.
Riemann integrals.
Types of convergence.
Product measures.
Signed measures.
The Radon-Nikodym theorem.
Differentiation.
Lp spaces.
Fourier transforms/
Riesz representation/.
Banach spaces.
Hilbert spaces.
Topology
Probability.
Harmonic functions.
Sobolev spaces.
Singular integrals.
Spectral theory.
Distributions.
Hints on exercises.