Department of Mathematics

Real Analysis Qualifier Topics List

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Last revised: July 2022

Preliminary background

We assume an undergraduate class in Advanced Calculus covering material similar to at least the first seven chapters of:
Walter Rudin, Principles of Mathematical Analysis, 3rd edition, McGraw-Hill, 1976.

Suggested Texts

1. Folland, Real Analysis, 2nd edition, Wiley 1999
2. Royden and Fitzpatrick, Real Analysis, 4th edition Prentice-Hall, 2010
3. Stein and Shakarchi, Real Analysis, Princeton, 2005
4. Makarov and Podkorytov, Real Analysis: Measures, Integrals and Applications, Springer, 2013
5. Terence Tao, An Introduction to Measure Theory, AMS, 2011
6. Terence Tao, An Epsilon of Room I, AMS, 2010

Qualifier Topics

1. Jordan measure and the Riemann-Darboux integral on R^d.
2. Lebesgue measure and the Lebesgue integral on R^d.
3. Boolean algebras and sigma-algebras of sets; the monotone class lemma.
4. Measure and integration on abstract measure spaces.
5. Littlewoods three principles, including Lusin and Egoroff.
6. Fatous lemma, the Monotone Convergence Theorem, and the Dominated Comvergence Theorem.
7. Relationships among modes of convergence; uniform integrability.
8. Pre-measures, outer measures, and the Hahn-Komogorov extension theorem.
9. Product measure spaces and the Fubini-Tonelli theorem, and applications.
10. The Lebesgue Differentiation Theorem in R^d.
11. Bounded variation, absolute continuity, and the Fundamental Theorems of Calculus for a.e. differentiable functions on R.
12. L^p-spaces.
13. Riesz representation theorem for positive linear functionals on continuos functions with compact support.
14. Elements of Functional Analysis: Banach spaces, Baire theorem, open mapping theorem, uniform boundedness principle, Hahn-Banach theorem; Hilbert spaces, orthonormal bases.
15. Linear functionals on Banach spaces, dual space, weak and weak* topologies, Banach-Alaouglu theorem.