Ph.D. Comprehensive Examination Topics
Real Analysis
2009

Set Operations and Algebras of Sets

Sequences, limsup, liminf

Open and closed sets of reals

Continuous functions

Borel sets

Lebesgue measure

Outer measure

Measurable sets

Measurable functions

Lusin's Theorem

Lebesgue integral and Differentation

Convergence theorems

Fatou's Lemma

Monotone Convergence Theorem

Lebesgue Dominated Convergence Theorem

Differentiation

Total Variation, Functions of Bounded Variation

Absolutely Continuous Funtions

Lebesgue Differentiation Theorem, Lebesgue Density of a Set

Classical Banach Spaces

L^p spaces, Hilbert Spaces

Minkowski and Hölder inequalities

Completeness

General Measure and Integration

Measure spaces, Carathodory measurable sets, measurable functions, convergence theorems

Signed measures, examples of measures

Radon-Nikodym Theorem

Lebesgue decomposition, singular, absolutely continuous

Product measures

Fubini's and Tonelli's Theorems

Modes of Convergence

Definitions of Pointwise (A.E.), Uniform, Mean L^p, in measure

Counterexamples

Relating various modes (Chapter 7 in Bartle, for example)

Egoroff's Theorem

References:

Real Analysis, H. Royden, Macmillan Publishing Co.
Measure and Integration, M. Munroe, Addison-Wesley Publishing Co.
Elements of Integration, Bartle, Wiley Classic Series
Real and Abstract Analysis, Measure Theory, integration and Hilbert Spaces, Stein and Shakarchi.

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