Course11110616179 lectures
Measure and Integration
Prof. S. Kesavan · IMSc
About
The theory of measure and integration is now an integral part of any masters or graduate program in mathemat ics in Indian Universities. It is an important prerequisite for most analysis based courses. In this course, we will start with a study of abstract measures with the Lebesgue measure being the most important example.
After looking at measurable functions and some types of convergences for such functions, we will introduce the notion of the Lebesgue integral in an abstract measure space and study its properties. We will relate the two important processes of the calculus – differentiation and integration – in the context of the Lebesgue integral. We will also study measures and integrals on product spaces and also signed measures. Finally we will study the important properties of L^p – spaces.
Intended audience: MSc (Mathematics) and above
Prerequisites: MSc Real Analysis, Topology, Linear AlgebraINTENDED AUDIENCE: MSc (Mathematics) and above
Lectures
Week 1
- 1.1 - Preamble
- 1.2 - Algebras of sets
- 1.3 - Measures on rings
- 1.4 - Outer-measure
- 1.5 - Measurable sets
Week 2
- 2.1 - Caratheodory's method
- 2.2 - Exercises
- 2.3 - Exercises
- 2.4 - Lebesgue measure: the ring
- 2.5 - Construction of the Lebesgue measure
- 2.6 - Errata
- 2.7 - The Cantor set
- 2.8 - Approximation
Week 3
- 3.1 - Approximation
- 3.2 - Approximation
- 3.3 - Translation Invariance
- 3.4 - Non-measurable sets
- 3.5 - Exercises
- 3.6 - Measurable functions
Week 4
- 4.1 - Measurable functions
- 4.2 - The Cantor function
- 4.3 - Exercises
- 4.4 - Egorov's theorem
- 4.5 - Convergence in measure
- 4.6 - Convergence in measure
Week 5
- 5.1 - Convergence in measure
- 5.2 - Exercises
- 5.3 - Integration: Simple functions
- 5.4 - Non-negative functions
- 5.5 - Monotone convergence theorem
- 5.6 - Examples
- 5.7 - Fatou's lemma
- 5.8 - Integrable functions
Week 6
- 6.1 - Dominated convergence theorem
- 6.2 - Dominated convergence theorem: Applications
- 6.3 - Absolute continuity
- 6.4 - Integration on the real line
- 6.5 - Examples
- 6.6 - Weierstrass' theorem
- 6.7 - Exercises
- 6.8 - Exercises
Week 7
- 7.1 - Vitali covering lemma
- 7.2 - Monotonic functions
- 7.3 - Functions of bounded variation
- 7.4 - Functions of bounded variation
- 7.5 - Functions of bounded variation
- 7.6 - Differentiation of an indefinite integral
- 7.7 - Absolute continuity
Week 8
- 8.1 - Exercises
- 8.2 - Product spaces
- 8.3 - Product spaces: measurable functions
- 8.4 - Product measure
- 8.5 - Fubini's theorem
- 8.6 - Examples
- 8.7 - Examples
Week 9
- 9.1 - Integration of radial functions
- 9.2 - Measure of the unit ball in N dimensions
- 9.3 -Exercises
- 9.4 - Signed measures
- 9.5 - Hahn and Jordan decompositions
- 9.6 - Upper,lower and totaal variations of a signed measure; Absolute continuity
- 9.7 - Absolute continuity
Week 10
- 10.1 - Radon-Nikodym theorem
- 10.2 - Radon-Nikodym theorem
- 10.3 - Exercises
- 10.4 - Lebesgue spaces
- 10.5 - Examples. Inclusion questions
- 10.6 - Convergence in L^p
Week 11
- 11.1 - Approximation
- 11.2 - Applications
- 11.3 - Duality
- 11.4 - Duality
- 11.5 - Convolutions
- 11.6 - Convolutions
Week 12
Syllabus and materials
At NPTEL, as the course publishes them.
Credits and licence
- Institution
- IMSc
- Department
- Mathematics
- Teacher
- Prof. S. Kesavan
- Level
- Graduate
- Language
- English
- Subjects
- Mathematics
- Source
- NPTEL
- Licence
- CC BY-NC-SA, as NPTEL states it. Non-commercial use only. NPTEL's terms
- Attribution
- NPTEL states: "Distributed under Creative Commons Attribution-ShareAlike / CC BY - NC - SA", beside a BY-NC-SA badge. Coordinated by IIT Madras.
- Who can watch
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