Math 540. Real Analysis I

1. Measures on the line

Abstract measure theory, outer measure, Lebesgue measure on the real line, measurable sets, Borel sets, Cantor sets and functions, non-measurable sets.
(Optional: Baire's category theorem.)

2. Measurable functions

Structure of measurable sets, approximation of measurable functions by simple functions, Littlewood’s three principles, Egorov and Lusin’s theorems.

3. Integration

Lebesgue theory of integration, convergence theorems (Monotone Convergence, Fatou’s Lemma, little Fubini, Dominated Convergence), comparison of the Riemann and Lebesgue integrals, modes of convergence, approximation of integrable functions by continuous functions, Fubini’s theorem for the plane.
(Optional: product measures, the general Fubini-Tonelli theorem, applications to probability, the convolution product.)

4. Differentiability

Functions of bounded variation (structure and differentiability), absolutely continuous functions, maximal functions, fundamental theorem of calculus.
(Optional: the Radon-Nikodym theorem.)

5. L^p spaces on intervals and l ^p spaces

Jensen’s inequality, Hölder and Minkowski’s inequalities, class of L^p functions, completeness, duals of L^p; spaces, inclusions of L^p spaces.

6. Hilbert spaces and Fourier series

Elementary Hilbert space theory, orthogonal projections, Riesz representation theorem, Bessel’s inequality, Riemann-Lebesgue lemma, Parseval’s identity, completeness of trigonometric spaces.

Optional topics are not required for the comp exam.

Textbooks used in past semesters:
G. B. Folland, Real Analysis, John Wiley & Sons.
H. L. Royden, Real Analysis (3rd edition), Prentice Hall.
W. Rudin, Real and Complex Analysis, McGraw-Hill.