Measure Theory

Paper Code: 
24DMAT702
Credits: 
6
Contact Hours: 
90.00
Max. Marks: 
100.00
Objective: 

This course will enable the students to-

  1. Explore the concept of the abstract measure theory, definition and main properties of the integral.
  2. Construct Lebesgue's measure on the real line and in n-dimensional Euclidean space.
  3. Learn the advanced directions of measure theory.

 

Course Outcomes: 

 Course

Learning outcomes

(at course level)

Learning and teaching strategies

Assessment

Strategies

Course Code

Course Title

 

 

 

 

 

 

 

24DMAT

702

 

Measure Theory

(Theory)

 

 

 

CO95: Explore the basic concepts of set theory and measurability to define a measure, Non-measurable sets.

CO96: Analyze the concept of measurable functions and convergence.

CO97: Explore the bounded measurable functions with its properties.

CO98: Diagnose Square Summable Function. Create Parseval's identity, Riesz-Fischer theorem.

CO99: Explore Lp-Spaces and the Hölder inequality, Minkowski inequalities.

CO100: Contribute effectively in course-specific interaction.

Approach in teaching:

Interactive Lectures, Discussion, Power Point Presentations, Informative videos

 

Learning activities for the students:

Self learning assignments, Effective questions, presentations, Assigned tasks

 

 

Quiz, Individual and group projects,

Open Book Test, Semester End Examination

 

 

 

 

Unit I: 
Algebras of set:
18.00

 Algebra and algebras of sets, Algebras generated by a class of subsets, Borel sets, Lebesgue measure of sets of real numbers, Measurability and measure of a set, Existence of non-measurable sets.

 

Unit II: 
Measurable functions:
18.00

 Realization of non-negative measurable functions as limit of an increasing sequence of simple functions, Structure of measurable functions, Convergence in measure, Egoroff's theorem.

 

Unit III: 
Lebesgue integral:
18.00

 Lebesgue integral of bounded measurable functions, Lebesgue theorem on the passage to the limit under the integral sign for bounded measurable functions.

 

Unit IV: 
Summable functions:
18.00

 Space of square summable functions. Fourier series and coefficients, Parseval's identity, Riesz-Fisher Theorem. 

 

Unit V: 
Lp-spaces and related theorems:
18.00

 Lp-spaces, Holder-Minkowski inequalities, Completeness of Lp-spaces.

 

Essential Readings: 
  • Shanti Narayan, A Course of Mathematical Analysis, S.Chand & Co.New Delhi, 2005.
  • T.M. Apostol, Mathematical Analysis, Narosa Publishing House, New Delhi, 2002.
  • Walter Rudin, Real and Complex Analysis, McGraw-Hill Education, 2017.

 

References: 
  • P.K. Jain and S.K. Kaushik, An Introduction to Real Analysis, S.Chand & Co, New Delhi, 2000. 
  • R.R. Goldberg, Real Analysis, Oxford and IBH Publishing Company, New Delhi, 1970.         
  • Halsey Royden, Patrick Fitzpatrick, Real Analysis, Pearson’s United States Edition, 2010.
  • G.F. Simmons, Introduction to Topology and Modern Analysis, McGraw-Hill Book, New Delhi, 2017.
  • G. De. Barra, Measure Theory and Integration, New Age International Private Limited, 2013.
  • S.K. Berberian, Measure and Integration, McMillan, New York, 1965.
  • I.K. Rana, An Introduction to Measure and Integration, Narosa Publishing House New Delhi, 2007.

e- RESOURCES

 

JOURNALS

 

 

Academic Year: 
2024-2025
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