### Math 540. Real Analysis Instructor Syllabus

**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*; spaces, inclusions of

^{p}*L*spaces.

^{p}**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 and Patrick Fitzpatrick, Real Analysis (4th edition), Pearson, 2010.

W. Rudin, Real and Complex Analysis, McGraw-Hill.

**(Revised 6/10/10)**