Real Analysis

Real Analysis Syllabus

For the Oral Qualifying Exam

  • 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.
  • LΡ 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, Chapters 1-7, 11-12.
  • Reed and Simon, Methods of Mathematical Physics I: Functional Analysis, Chapters 1-2.
  • G.B. Folland, Real Analysis, Chapters 0-3, 6.