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
 

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