Real Analysis (Ph.D. and Ed.D.)

Preparatory Courses: Math 5143, 5153

1.

Algebras and sigma-algebras of sets, outer measures and the Caratheodory construction of measures, especially for Lebesgue-Stieltjes measures, Borel sets, Borel measures, regularity properties of measures, measurable functions.

2.

Construction of the integral with respect to a measure, convergence theorems: Lebesgue dominated convergence theorem, Fatou's Lemma, and monotone convergence theorem, Egorov's Theorem, Lusin's Theorem, product measures and Fubini's Theorem.

3.

Signed measures and the Hahn decomposition theorem, Radon-Nikodym Theorem, Lebesgue decomposition of a measure with respect to another measure, functions of bounded variation, absolutely continuous functions, Lebesgue-Stieltjes integrals.

4.

Topology on metric spaces and locally compact Hausdorff spaces, nets, Urysohn's Lemma, Tychonoff, Stone-Weierstrass, and Ascoli Theorems.

5.

Introductory functional analysis: Baire Category, Hahn-Banach theorem, uniform boundedness principle (Banach-Steinhaus), open mapping theorem, closed graph theorem, weak topologies, $Lp$ spaces, completeness of the $L1$ spaces, Minkowski and Holder inequalities, elementary Hilbert space theory, Fourier series in $L2$, Riesz Representation theorems in $Lp$ and C(X).

REFERENCES: Folland, Real Analysis ; Royden, Real Analysis ; Rudin, Real and Complex Analysis ; Hewitt and Stromberg, Real and Abstract Analysis .