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source: Bilkent Online Courses 2014年8月16日
MATH-501 Real Analysis - I (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 01 Category 50:58
Lecture 02 Borel sets 51:09
Lecture 03 Baire functions 52:05
Lecture 04 Concept of measure 47:47
Lecture 05 Measurable sets 51:36
Lecture 06 Lebesgue measure 49:59
Lecture 07 Approximation of measurable sets 50:29
Lecture 08 Lebesgue density theorem 50:55
Lecture 09 Hausdorff measures 51:00
Lecture 10 Extension of premeasures 52:07
Lecture 11 Nonmeasurable sets 49:36
Lecture 12 Measurable functions 48:09
Lecture 13 Review of mid-term exam 47:57
Lecture 14 Almost uniform convergence 48:51
Lecture 15 Egorovs theorem 49:38
Lecture 16 Lusin theorem 51:09
Lecture 17 Convergence in measure 50:58
Lecture 18 Lebesgue integral for bounded functions 50:05
Lecture 19 Monotone Convergence Theorem 50:38
Lecture 20 Fatou Lemma 48:51
Lecture 21 Lebesgue Dominated Convergence Theorem 50:22
Lecture 22 Characterizations of Integrability 50:33
Lecture 23 Indefinite Lebesgue Integral 51:35
Lecture 24 Differentiation of Monotone Function 51:51
Lecture 25 Indefinite Lebesgue Integral 49:45
Lecture 26 Absolutely Continuous Functions 48:48
Lecture 27 Signed Measures 53:27
Lecture 28 Hahn Decomposition 50:54
Lecture 29 Radon-Nikodym Theorem 50:41
Lecture 30 Product Measures 53:05
Lecture 31 Fubini Theorem 46:10
Lecture 32 Applications of Fubini Theorem 48:50
Lecture 33 Spaces of Integrable Functions 51:38
Lecture 34 Rearrangement of Functions 50:33
Lecture 35 Approximation in LP 49:49
Lecture 36 Riesz' Representation Theorem 52:14
Lecture 37 Hielbert Spaces 1:19:28
