Measure Theory - IMSc

By Prof. Indrava Roy   |   IMSC
Learners enrolled: 475
One of the main goals of Lebesgue's measure theory is to develop a fundamental tool for carrying out integration which behaves well with taking limits, and admitting a vast class of functions for which Riemann's integration theory is not applicable. Even though the crux of measure theory was to produce a good integration theory, it turns out that it also gives new ways of thinking about “measuring” objects, which is very useful for many other areas of mathematics such as probability theory as well as more advanced topics like harmonic analysis, ergodic theory, etc. Real-world applications of measure theory can be found in physics, economics, finance etc. Measure theoretic techniques are thus a must-have for any mathematician.

INTENDED AUDIENCE
1st year M.Sc. onwards
PREREQUISITES Set theory and Basic real analysis
INDUSTRIES  SUPPORT       : None
Summary
 Course Status : Completed Course Type : Core Duration : 12 weeks Category : Mathematics Credit Points : 3 Level : Postgraduate Start Date : 26 Jul 2021 End Date : 15 Oct 2021 Enrollment Ends : 09 Aug 2021 Exam Date : 24 Oct 2021 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:Introduction and Motivation of Measure theory, Jordan measurability and Jordan content
Week 2:Basic properties of Jordan content and connection with Riemann integrals, Motivation and definition of Lebesgue outer measure on R^n
Week 3: Properties of Lebesgue outer measure on R^n, Caratheodory extension theorem
Week 4:Lebesgue measurability, Vitali and Cantor sets, Boolean and sigma algebras
Week 5:Abstract measure spaces with examples: Borel and Radon measures, Metric outer measures, Lebesgue-Stieljes measures, Hausdorff measures and dimension* (extra content)
Week 6:Measurable functions and abstract Lebesgue integration, Monotone convergence theorem, Fatou's lemma, Tonnelli's theorem
Week 7: Borel-Cantelli Lemma, Dominated convergence theorem, the space L^1
Week 8:Various modes of convergence and their inter-dependence
Week 9:Riesz representation theorem, examples of measures constructed via RRT
Week 10:Product measures and Fubini-Tonnelli theorem
Week 11: Hardy-Littlewood Maximal inequality and Lebesgue's differentiation theorem
Week 12:Lebesgue's differentiation theorem (continued)

Books and references

1. Introduction to Measure theory, Terence Tao (Freely available online from: https://terrytao.files.wordpress.com/2011/01/measure-book1.pdf )
2. Real Analysis, Measure Theory, Integration, and.Hilbert Spaces. Elias M. Stein &. Rami Shakarchi, Princeton University Press
3. Real analysis: modern techniques and their application, G. B. Folland, Wiley.

Prof. Indrava Roy

IMSC
Indrava Roy is an Assistant Professor of Mathematics at the Institute of Mathematical Sciences, Chennai

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: 24 October 2021 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