Math 5111 : Real Analysis (Measure and Integration) Spring 2012 *`Alexander Teplyaev`*

course time and room: TuTh 9:30-10:45am, MSB 415

The course is open for all graduate students who passed 5110, and to other students by permission; the course can be repeated either as an audit or as an independent study (don't hesitate to ask for permission if needed).

office: MSB M222 -- office hours: check this link phone: (860)486-3206 email: teplyaevmath.uconn.edu http://www.math.uconn.edu/~teplyaev

`homework`

Textbook

Real Analysis for Graduate Students: Measure and Integration Theory by Richard Bass (special note: you are not allowed to print books on departmental printers)

Course description

5111 --- Abstract integration: Lebesgue integration theory, outer measures and Caratheodory's theorem, Fatou's lemma, monotone and dominated convergence theorems. Measure theory: positive Borel measures, Riesz representation theorem for positive linear functionals on C(K), complex measures, Hahn-Jordan and Lebesgue decompositions, Radon-Nykodim theorem and differentiation of measures, Riemann-Stieltjes integral, the Lebesgue measure on R^d. L^p spaces: Cauchy-Bunyakovsky-Schwarz, Hoelder, Minkowski and Jensen inequalities, L² and L^p spaces as Hilbert and Banach spaces, Riesz representation theorem for bounded functionals on L^p. Integration on product spaces, Fubini's and Tonelli's theorems, Fourier transform and Plancherel theorem. See also Real Analysis Prelim Study Guide. Textbook choices: Real Analysis for Graduate Students: Measure and Integration Theory by Rich Bass. Real and Complex Analysis by W. Rudin. Real Analysis and Probability by R.M. Dudley. Real Analysis: Modern Techniques and Their Applications by G. Folland. Supplementary reading: Real Analysis by H. Royden and P. Fitzpatrick. Real Analysis: Measure Theory, Integration, and Hilbert Spaces by E.M. Stein and R. Shakarchi. Measure and Integral by R.L. Wheeden and A. Zygmund.

http://www.math.uconn.edu