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.

L^p 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.

References:

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.