# Measure and Integration

By Prof. S. Kesavan   |   IMSc
Learners enrolled: 400
About the course: 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.

PRE-REQUISITES: MSc Real Analysis,  Topology, Linear Algebra

INTENDED AUDIENCE: MSc (Mathematics) and above
Summary
 Course Status : Completed Course Type : Core Duration : 12 weeks Category : Mathematics Credit Points : 3 Level : Postgraduate Start Date : 25 Jul 2022 End Date : 14 Oct 2022 Enrollment Ends : 08 Aug 2022 Exam Date : 30 Oct 2022 IST

Note: This exam date is subjected to change based on seat availability. You can check final exam date on your hall ticket.

### Course layout

Week 1: Motivation, abstract measures, Caratheodory’s method of extension, completion of a measure.
Week 2: Construction of the Lebesgue meaure, approximation properties.
Week 3: Translation invariance, nonmeasurable sets, measurable functions.
Week 4: Properties of measurable functions, Cantor function.
Week 5: Convergence, Egorov’s theorem, convergence in measure.
Week 6: Lebesgue integration, convergence theorems.
Week 7: Comparison with the Riemann integral, some applications (Weierstrass’ theorem).
Week 8: Differentiation: Monotone functions, functions of bounded variation, absolute continuity.
Week 9: Product spaces, Fubini’s theorem.
Week 10: Signed measures, Radon-Nikodym theorem.
Week 11: L^p-Spaces: Basic properties, approximation, applications.
Week 12: Duality, convolutions.

### Books and references

S. Kesavan, Measure and Integration, TRIM Series No.77, Hindustan Book Agency.

### Prof. S. Kesavan

IMSc
S. Kesavan retired as Professor from the Institute of Mathematical Sciences, Chennai. He obtained his doctoral degree from the Universite de Pierre et Marie Curie (Paris VI), France. His research interests are in Partial Differential Equations. He is the author of five books. He is a Fellow of the Indian Academy of Sciences and the National Academy of Sciences, India. He has served as Deputy Director of the Chennai Mathematical Institute (2007-2010) and two terms (2011-14, 2015-18) as Secretary (Grants Selection) of the Commission for Developing Countries of the International Mathematical Union. He was a member of the National Board for Higher Mathematics during 2000-2019.

### Course certificate

The course is free to enroll and learn from. But if you want a certificate, you have to register and write the proctored exam conducted by us in person at any of the designated exam centres.
The exam is optional for a fee of Rs 1000/- (Rupees one thousand only).
Date and Time of Exams: 30 October 2022 Morning session 9am to 12 noon; Afternoon Session 2pm to 5pm.
Registration url: Announcements will be made when the registration form is open for registrations.
The online registration form has to be filled and the certification exam fee needs to be paid. More details will be made available when the exam registration form is published. If there are any changes, it will be mentioned then.
Please check the form for more details on the cities where the exams will be held, the conditions you agree to when you fill the form etc.

CRITERIA TO GET A CERTIFICATE

Average assignment score = 25% of average of best 8 assignments out of the total 12 assignments given in the course.
Exam score = 75% of the proctored certification exam score out of 100

Final score = Average assignment score + Exam score

YOU WILL BE ELIGIBLE FOR A CERTIFICATE ONLY IF AVERAGE ASSIGNMENT SCORE >=10/25 AND EXAM SCORE >= 30/75. If one of the 2 criteria is not met, you will not get the certificate even if the Final score >= 40/100.

Certificate will have your name, photograph and the score in the final exam with the breakup.It will have the logos of NPTEL and IIT Madras. It will be e-verifiable at nptel.ac.in/noc.

Only the e-certificate will be made available. Hard copies will not be dispatched.

Once again, thanks for your interest in our online courses and certification. Happy learning.

- NPTEL team