MATH-501 Real Analysis - I
Assoc. Prof. Dr. Alexandre Gontcharov
2009-2010- Fall
Concepts of integration. Henstock-Kurzweil integral. Borel sets, Bair functions. Outer measures. Measurable sets. Lebesgue and Lebesgue-Stieltjes measures. Lebesgue density theorem. Hausdorff measures and Hausdorff dimension. Measurable functions. Lusin’s and Egorov’s theorems. Convergence in measure. Lebesgue integral. Basic theorems of Lebesgue integral. Modes of convergence. Differentiation of indefinite Lebesgue integral. Signed measures. The Radon- Nikodym theorem. Product measures. Spaces of integrable functions.
 | Lecture 37 (2009-12-29) Hiebert Spaces | |
 | Lecture 36 (2009-12-28) Riesz' Representation Theorem | |
 | Lecture 35 (2009-12-22) Approximation in LP | |
 | Lecture 34 (2009-12-22) Rearrangement of Functions | |
 | Lecture 33 (2009-12-21) Spaces of Integrable Functions | |
 | Lecture 32 (2009-12-15) Applications of Fubini Theorem | |
 | Lecture 31 (2009-12-15) Fubini Theorem | |
 | Lecture 30 (2009-12-14) Product Measures | |
 | Lecture 29 (2009-12-08) Radon-Nikodym Theorem | |
 | Lecture 28 (2009-12-08) Hahn Decomposition | |
 | Lecture 27 (2009-12-07) Signed Measures | |
 | Lecture 26 (2009-12-01) Absolutely Continuous Functions | |
 | Lecture 25 (2009-12-01) Indefinite Lebesgue Integral | |
 | Lecture 24 (2009-11-17) Differentiation of Monotone Function | |
 | Lecture 23 (2009-11-17) Indefinite Lebesgue Integral | |
 | Lecture 22 (2009-11-16) Characterizations of Integrability | |
 | Lecture 21 (2009-11-10) Lebesgue Dominated Convergence Theorem | |
 | Lecture 20 (2009-11-10) Fatou Lemma | |
 | Lecture 19 (2009-11-09) Monotone Convergence Theorem | |
 | Lecture 18 (2009-11-03) Lebesgue integral for bounded functions | |
 | Lecture 17 (2009-11-03) Convergence in measure | |
 | Lecture 16 (2009-11-02) Lusin theorem | |
 | Lecture 15 (2009-10-27) Egorovs theorem | |
 | Lecture 14 (2009-10-27) Almost uniform convergence | |
 | Lecture 13 (2009-10-26) Review of mid-term exam | |
 | Lecture 12 (2009-10-20) Measurable functions | |
 | Lecture 11 (2009-10-20) Nonmeasurable sets | |
 | Lecture 10 (2009-10-19) Extension of premeasures | |
 | Lecture 9 (2009-10-13) Hausdorff measures | |
 | Lecture 8 (2009-10-13) Lebesgue density theorem | |
 | Lecture 7 (2009-10-12) Approximation of measurable sets | |
 | Lecture 6 (2009-10-06) Lebesgue measure | |
 | Lecture 5 (2009-10-06) Measurable sets | |
 | Lecture 4 (2009-10-05) Concept of measure | |
 | Lecture 3 (2009-09-29) Baire functions | |
 | Lecture 2 (2009-09-29) Borel sets | |
 | Lecture 1 (2009-09-28) Category |