Analysis III: Measure and Integration Theory (MAT 425)

List of (big) theorems and topics: • Abstract measure theory. • The Lebesgue integral: see Section 2.7. • Radon-Nikodym derivative: see Section 5. • Fubini, dominated convergence (see Theorem 2.61), monotone convergence (see Theorem 2.47), Fatou (see Lemma 2.53). • Borel-Cantelli appears in HW3Q9. • Ergodic theorems. • Carathéodory’s theorem (see Theorem 2.70 and Theorem 2.76). • The Lebesgue-Stieltjes integral appears in HW3Q5. • Tempered distributions. • Hilbert space theory and applications to Fourier Transforms, and partial differential equations. • Some probability theory • Introduction to fractals