Measure Theory

By Prof. Indrava Roy | IMSC

Learners enrolled: 1235

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
Language for course content :	English
Duration :	12 weeks
Category :	Mathematics
Credit Points :	3
Level :	Postgraduate
Start Date :	14 Sep 2020
End Date :	04 Dec 2020
Enrollment Ends :	25 Sep 2020
Exam Date :	19 Dec 2020 IST

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

Page Visits

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.

Instructor bio

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:19 December 2020, 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.

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