Advanced Real Analysis I

Course Description

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.

Advanced Real Analysis I
Fubini's Theorem (studied in great detail in Lecture 31) is usually used to calculate the volume of three dimensional bodies.
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Video Lectures & Study Materials

Visit the official course website for more study materials: http://video.bilkent.edu.tr/course_videos.php?courseid=12

# Lecture Play Lecture
1 Category (50:58) Play Video
2 Borel Sets (51:09) Play Video
3 Baire Functions (52:05) Play Video
4 Concept of Measure (47:47) Play Video
5 Measurable Sets (51:36) Play Video
6 Lebesgue Measure (49:59) Play Video
7 Approximation of Measurable Sets (50:29) Play Video
8 Lebesgue Density Theorem (50:55) Play Video
9 Hausdorff Measures (51:00) Play Video
10 Extension of Premeasures (52:07) Play Video
11 Nonmeasurable Sets (49:36) Play Video
12 Measurable Functions (48:09) Play Video
13 Review of Midterm Exam (47:57) Play Video
14 Almost Uniform Convergence (48:51) Play Video
15 Egorov's Theorem (49:38) Play Video
16 Lusin's Theorem (51:09) Play Video
17 Convergence in Measure (50:58) Play Video
18 Lebesgue Integral for Bounded Functions (50:05) Play Video
19 Monotone Convergence Theorem (50:38) Play Video
20 Fatou's Lemma (48:51) Play Video
21 Lebesgue's Dominated Convergence Theorem (50:22) Play Video
22 Characterizations of Integrability (50:33) Play Video
23 Indefinite Lebesgue Integral (51:35) Play Video
24 Differentiation of Monotone Function (51:51) Play Video
25 Indefinite Lebesgue Integral (49:45) Play Video
26 Absolutely Continuous Functions (48:48) Play Video
27 Signed Measures (53:27) Play Video
28 Hahn Decomposition Theorem (50:54) Play Video
29 Radon-Nikodym Theorem (50:41) Play Video
30 Product Measures (53:05) Play Video
31 Fubini's Theorem (46:10) Play Video
32 Applications of Fubini's Theorem (48:50) Play Video
33 Spaces of Integrable Functions (51:38) Play Video
34 Rearrangement of Functions (50:33) Play Video
35 Approximation in LP (49:49) Play Video
36 Riesz Representation Theorem (52:14) Play Video
37 Introduction to Hilbert Spaces (1:19:28) Play Video

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