# Measure Theory - IITB

By Prof. I. K. Rana   |   IIT Bombay
Learners enrolled: 497
This is a course on the concepts of Measure and Integration. Normally, this is a core course for M,.Sc. Mathematics and Statistics students. The concepts find applications in advance Analysis Courses, Signal Processing, Financial Mathematics courses.
INTENDED AUDIENCE : B.Tech Dual degree in Electrical, M.Sc. Physics, mathematics
PREREQUISITES : Basic Course in Real Analysis
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 : 23 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

Lecture 1A Introduction, Extended Real Numbers
Lecture 1B Introduction, Extended Real Numbers
Lecture 2A Algebra and Sigma Algebra of Subsets of a Set
Lecture 2B Algebra and Sigma Algebra of Subsets of a Set
Lecture 3A Sigma Algebra generated by a Class
Lecture 3B Sigma Algebra generated by a Class

Week 2

Lecture 4A Monotone Class
Lecture 4B Monotone Class
Lecture 5A Set Functions
Lecture 5B Set Functions
Lecture 6A The Length Function and its Properties
Lecture 6B The Length Function and its Properties

Week 3

Lecture 7A Countably Additive Set Functions on Intervals
Lecture 7B Countably Additive Set Functions on Intervals
Lecture 8A Uniqueness Problem for Measure
Lecture 8B Uniqueness Problem for Measure

Week 4

Lecture 9A Extension of Measure
Lecture 9B Extension of Measure
Lecture 10A Outer Measure and its Properties
Lecture 10B Outer Measure and its Properties
Lecture 11A Measurable Sets
Lecture 11B Measurable Sets

Week 5

Lecture 12A Lebesgue Measure and its Properties
Lecture 12B Lebesgue Measure and its Properties
Lecture 13A Characterization of Lebesgue Measurable Sets
Lecture 13B Characterization of Lebesgue Measurable Sets

Week 6

Lecture 14A Measurable Functions
Lecture 14B Measurable Functions
Lecture 15A Properties of Measurable Functions
Lecture 15B Properties of Measurable Functions
Lecture 16A Measurable Functions on Measure Spaces
Lecture 16B Measurable Functions on Measure Spaces

Week 7

Lecture 17A Integral of Nonnegative Simple Measurable Functions
Lecture 17B Integral of Nonnegative Simple Measurable Functions
Lecture 18A Properties of Nonnegative Simple Measurable Functions
Lecture 18B Properties of Nonnegative Simple Measurable Functions
Lecture 19A Monotone Convergence Theorem and Fatou's Lemma
Lecture 19B Monotone Convergence Theorem and Fatou's Lemma

Week 8

Lecture 20A Properties of Integrable Functions and Dominated Convergence Theorem
Lecture 20B Properties of Integrable Functions and Dominated Convergence Theorem
Lecture 21A Dominated Convergence Theorem and Applications
Lecture 21B Dominated Convergence Theorem and Applications

Week 9

Lecture 22A Lebesgue Integral and its Properties
Lecture 22B Lebesgue Integral and its Properties
Lecture 23A Product Measure, an Introduction
Lecture 23B Product Measure, an Introduction
Lecture 24A Construction of Product Measures
Lecture 24B Construction of Product Measures

Week 10

Lecture 25A Computation of Product Measure - I
Lecture 25B Computation of Product Measure - I
Lecture 26A Computation of Product Measure - II
Lecture 26B Computation of Product Measure - II

Week 11

Lecture 27A Integration on Product Spaces
Lecture 27B Integration on Product Spaces
Lecture 28A Fubini's Theorems
Lecture 28B Fubini's Theorems

Week 12

Lecture 29A Lebesgue Measure and Integral on R2
Lecture 29B Lebesgue Measure and Integral on R2
Lecture 30A Properties of Lebesgue Measure on R2
Lecture 30B Properties of Lebesgue Measure on R2
Lecture 31A Lebesgue Integral on R2
Lecture 31B Lebesgue Integral on R2

### Books and references

An introduction to Measure and Integration – Inder K Rana, Narosa Publishers

### Prof. I. K. Rana

IIT Bombay
I am Emeritus Fellow at Department of Mathematics, IIT Bombay. I have taught couses at B.Tech, M.Sc., and Ph.D level for the last 34 years.

### 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: 23 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 Bombay. 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