### MATH 5111: Real Analysis

**Description:** 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,

*L*

^{2}and

*L*spaces as Hilbert and Banach spaces, Riesz representation theorem for bounded functionals on

^{p}*L*. Integration on product spaces, Fubini's and Tonelli's theorems, Fourier transform and Plancherel theorem. For prelim preparation, see the prelim study guide.

^{p}**Prerequisites:** MATH 5110.

**Offered:** Spring

**Credits:** 3

### MATH 5111 - Section 1: Measure Theory

**Description:** 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 Rd. Lp spaces: Cauchy-Bunyakovsky-Schwarz, Hoelder, Minkowski and Jensen inequalities, L2 and Lp spaces as Hilbert and Banach spaces, Riesz representation theorem for bounded functionals on Lp. Integration on product spaces, Fubini's and Tonelli's theorems, Fourier transform and Plancherel theorem.

**Credits:** 3

**Sections: **Spring 2016 on Storrs Campus

PSCourseID | Course | Sec | Comp | Time | Room | Instructor |
---|---|---|---|---|---|---|

02501 | 5111 | 001 | Lecture | TuTh 02:00:00 PM-03:15:00 PM | MSB203 | Chousionis, Vasilis |