INDIAN STATISTICAL INSTITUTE (Delhi Centre) 7 S. J. S. Sansanwal Marg, New Delhi 110016

Instructor: Antar Bandyopadhyay (E-mail: antar (at) isid (dot) ac (dot) in)
Office: Room # 208 on the first floor of the Faculty Block.

Teaching Assistant (TA): Md Amir Hossain (E-mail: mdamirhossain18 (at) gmail (at) com)
TA's Office Hours: TBA

Class Time: Monday 11:00 AM - 01:00 PM & Wednesday: 10:00 AM - 12:00 Noon (2 x 120 minutes per week)
Lecture Hall: Room # 24 Academic Block (First Floor).
Vertual Google Classroom (for announcement purpose only)

Course Duration (including examinations): July 20 - November 20, 2026
(Total of 18 Weeks = 7 Weeks of Classes + 1 Midterm Examination Week + 7 Weeks of Classes + 3 Weeks of Study Leave and Final Examinations).

Midterm Examination:
Date: TBA (during September 07 - 11, 2026)
Duration: TBA (two hours)
Venue: TBA

Final Examination:
Date: TBA (during November 02 - 20, 2026)
Duration: TBA (three hours)
Venue: TBA

Course Outline:

• Review of the notion of cardinality of sets, finite, countable, uncountable sets. [1 week]
• Algebra of sets, σ-algebra, examples, Borel σ-algebra. Countable-co-countable σ-algebra. Countably generated σ-algebra. [1 week]
• Monotone classes and the Monotone Class Theorem, Good Set Principle. [1 weeks]
• Measurable functions, sum, product, maximum, minimum of measurable functions. Limits of measurable functions. Simple functions, Monotone Approximation Theorem. [1 week]
• Definition of measure, set functions, finite and countable additivity. Finite and infinite measures, probability measures, basic laws of measures. σ-finite measures. [1/2 week]
• π-systems, λ-systems. Dinkin's π-λ Theorem. Uniqueness theorems. [1/2 week]
• Existence and extension theorems. Outer measure and Carathéodory's Extension Theorem. [0 week: done through homework assignments]
• Construction of the Lebesgue measure on unit interval, real line and d-dimensional Euclidean space. Properties of the Lebesgue measure. [1 week]
• Lebesgue Theory of Integration. Monotone Convergence Theorem (MCT), Fatou's Lemma, Dominated Convergence Theorem (DCT). [1 and 1/2 weeks]
• Calculus of measure zero sets. Concept of "almost surely (a.s)". Change of variable formula. [1/2 week]
• Product space, basic definitions, sections. Product measure, existence and uniqueness. Fubini's Theorem, applications. [1 and 1/2 weeks]
• L_p-spaces and inequalities. Riesz-Fischer Theorem, approximation by step functions and continuous functions. [1 and 1/2 weeks]
• Absolute continuity and singularity of measures. Signed measures, Hahn-Jordon decomposition. [1 week]
• Radon-Nikodym Theorem, Lebesgue decomposition. Properties of the Radon-Nikodym derivative. [1 week]
• Functions of Bounded Variation. [1/2 week]
• Complex Measures and their properties. [1/2 week]

References:

• H. L. Royden and Patrick Fitzpatrick: Real Analysis, Pearson, 4th edition.
• Robert B. Ash and Catherine A. Doleans-Dade: Probability and measure theory, GTM(211), Academic Press, 2nd edition.
• Elias M. Stein and Rami Shakarchi: Real Analysis: Measure Theory, Integration and Hilbert Spaces, Princeton Lectures in Analysis.
• Gerald B. Folland: Real Analysis: Modern Techniques and Their Applications, Pure and Applied Mathematics, A Wiley Series.
• G. de Barra: Measure Theory and Integration, Woodhead Publishing Limited.

Prerequisites/Expected:

• Real Analysis (at the level of Principles of Mathematical Analysis, W. Rudin).
• Liner Algebra (at the level of Finite Dimensional Vector Spaces, P. R. Halmos).

Grading Policy:

• Assignments: 10% of the total credit.
• Quizzes: 10% of the total credit.
• Midterm Exam: 25% of the total credit.
• Final Exam: 55% of the total credit.

Assignment Policies:

• There will be a total of 14 sets of homework assignments each carrying a total of 10 points. The average of the 10 best assignment scores will be taken for the Assignment component of the final grading.
• The assignments will be uploaded in the Google Classroom on every Wednesday,starting from July 22, 2026. Each assignment will be due on the following Wednesday in the start of the class (that is by 10:00 AM). Assignments have to to be submitted in hard copy in your own handwritten solutions. For example the first assignment is due on Wednesday, July 29, 2026.
• Each Assignment will mostly be based on the course materials which will be covered in the class in the week of its assignment. Occasionally they may introduce/augment the discussions and also may have problems on topics which has been covered in some earlier week.
• Late submission of an assignment will NOT BE ACCEPTED. If you can not submit an assignment on time, don't worry about it and try to do well in the others. It will not count in your final grade since you have four extra assignments anyway.
• Assignments will be graded by the TA and the Graded assignments will be returned in the class in the week following its submission.

Quiz Policies:

• There will be at least four quizzes as surprise tests given in the class. This means there shall be no pre-scheduling. A quiz will cover materials done in the lectures given in the weeks prior to it.
• Each quiz will be of 10 points and will be of 15 - 20 minutes duration.
• Final grade for the quizzes will be (Best Score Before Midterm + Best Score Between Midterm and Final)/2.
• There will be NO supplementary quiz given for any student who may miss a quiz for whatsoever reason. If you miss one then do not worry, try doing well in the others.
• All quizzes will be part of the final grading.
• All quizzes will be closed note and closed book examinations.

Exam Policies:

• The Midterm and the Final Examinations will be open notes examinations. That means, students will be allowed to bring his/her own hand written notes, study materials, list of theorems etc. But no printed or photocopied materials will be allowed.
• Any unfair means used by any students in the examinations (including the quizzes) will be dealt with the strictest possible measures, as per the Institute rules.

Regrading Policy:

• Regrading of homeworks or exams will only be undertaken in cases where, you believe there has been a genuine error or misunderstanding. Please note that our primary aim in grading is consistency, so that all students are treated the same; for this reason, we will not adjust the score of one student on an issue of partial credit, unless the score allocated clearly deviates from the grading policy we adopted for that problem.
• If you wish to request a regrading of a homework or exam, you must return it to the instructor with a written note on a separate piece of paper explaining the problem.
• The entire assignment or the exam may be regraded, so be sure to check the solutions to ensure that your overall score will go up after regrading.
• All such requests must be received within one week from the date on which the homework or exam was made available for return.

Last modified July 19, 2026.