The Syllabus for the Qualifying Examination in Real Analysis

Outer measure, measurable sets, sigma-algebras, Borel sets, measurable functions, the Cantor set and function, nonmeasurable sets.

Lebesgue integration, Fatou's Lemma, the Monotone Convergence Theorem, the Lebesgue Dominated Convergence Theorem, convergence in measure.

Lp spaces, Hoelder and Minkowski inequalities, completeness, dual spaces.

Abstract measure spaces and integration, signed measures, the Hahn decomposition, the Radon-Nikodym Theorem, the Lebesgue Decomposition Theorem.

Product measures, the Fubini and Tonelli Theorems, Lebesgue measure on real n-space.

Equicontinuous families, the Ascoli-Arzela Theorem.

Hilbert spaces, orthogonal complements, representation of linear functionals, orthonormal bases.


H. L. Royden, Real Analysis, Chap. 1 - 7, 11, 12.
M. Reed and B. Simon, Methods of Mathematical Physics I: Functional Analysis, chapters one and two.
G. B. Folland, Real Analysis, Chap. 0 - 3, 6.