This is a Qualifying Eligible (QE) course for the Math PhD with regular, graded HW and a comprehensive final exam.

#### Prerequisites

Basic multivariable analysis (the equivalent of Math 531 and 532).

#### Syllabus

- 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. - Expectation, variance, and conditional expectation.
- 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.
- Fourier series and Fourier transform. Parseval’s identity.

#### 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.