Measure Theory
Fall 2011
Instructor: Antar Bandyopadhyay
(Email: antar (at) isid (dot) ac (dot) in
Office: 208 Faculty Building).
Class Time: Mon Thu 14:00 - 16:00 in Class Room 24.
Instructor's Office Hours: Mon 16:00 - 17:30
Course Duration: August 11 - December 1, 2011
Midterm Examination: October 3, 2011 (Monday), Time: 14:00 - 16:00 hours
Final Examination: December 8, 2011 (Thursday), Time: 10:00 - 13:00 hours
Course Outline:
- Basic notion of cardinality of sets, finite, countable, uncountable sets.
- Measure Zero sets. Construction of Lebesgue measure on [0,1] interval, outer/exterior measure.
Properties of Lebesgue measure.
- Introduction to abstract measure theory, algebra of sets, σ-algebra, examples, Borel σ-algebra.
- Definition of measure, set functions, finite and countable additivity.
Finite and infinite measures, probability measures.
- Measurable functions, sum, product, maximum, minimum of measurable
functions. Limits of measurable functions. Random variables,
simple functions, Monotone Approximation Theorem.
- Monotone classes, π-systems, λ-systems. Monotone Class Theorem
and Dinkin's π-λ Theorem. Uniqueness theorems.
- Existence and extension theorems. Outer measure,
Carathéo Dory's Extension Theorem.
Construction of the Lebesgue measure on d-dimensional Euclidean space.
Properties of the Lebesgue measure.
- Lebesgue Theory of Integration. MCT, Fatou's Lemma, DCT,
Calculus of measure zero sets. Concept of
"almost surely (a.s)". Change of variable formula.
- Product space, basic definitions, sections. Product measure, existence
and uniqueness. Fubini's Theorem, applications.
- Definition of signed measures, Positive and negative sets. Hahn-Jordan Decomposition.
Absolute continuity of two σ-finite measures. Radon-Nikodyme Theorem and Lebesgue Decomposition.
- Regularity properties of Lebesgue measures on d-dimensional Euclidean space.
References:
- Real Analysis by E. M. Stein and R. Shakarchi.
- Real Analysis by H. L. Royden.
- Real and Complex Analysis by W. Rudin.
- Measure Theory by P. R. Halmos.
- Real Analysis and Probability by R. B. Ash.
- Measure and Probability by S. Athreya and V. S. Sunder.
- Probability and Measure by P. Billingsley.
- Probability: Theory and Examples by R. Durrett.
- Introduction to Probability and Measure by
K. R. Parthasarathy.
Prerequisites:
- Real Analysis (at the level of Principles of Mathematical Analysis by W. Rudin).
- Liner Algebra (at the level of Finite Dimensional Vector Spaces by P. R. Halmos).
Grading Policy:
- Assignments: 20% of the total credit.
- Quizzes: 10% of the total credit.
- Midterm Exam: 20% of the total credit.
- Final Exam: 50% of the total credit.
Assignment Policies:
- There will be a total of 6 sets of homework assignments each
carrying a total of 40 points. 5 best assignment scores
will be taken for the final grading.
- The assignments will be given in class on every alternate Monday,
and it will be due in class on Thursday of the following week. Each assignment
set will be
based on course materials covered in the lectures in the week before and the week it is assigned.
For example, the assignment given on August 29, 2011 (Monday)
will be due on September 8, 2011 (Thursday).
- 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 few extra assignments anyway.
- Graded assignments will be returned in the class on the next
Monday following the due date. For example, the assignment which is due on September 8, 2011 (Thursday)
will be returned after grading on September 12, 2011 (Monday).
- Click here
for downloading the assignments.
Quiz Policies:
- There will be some number of quizzes as surprise tests given in the
class. This means there shall be no pre-scheduling. A quiz will cover
all the materials done in the lectures prior to it.
- 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 other.
- Each quiz will be a closed books, closed notes test.
Exam Policies:
- Each examination (except for the quizzes) will be
an open note examination. That means, students are allowed
to bring his/her own hand written notes, study materials, list of
theorems etc.
Last modified November 2, 2011.