Measure Theory

Paper Code: 
24MAT122
Credits: 
5
Contact Hours: 
75.00
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

24MAT

122

Measure Theory

(Theory)

 

CO7: Explore the basic concepts of set theory and measurability to define a measure, Non-measurable sets.

CO8: Analyze the concept of measurable functions and convergence.

CO9: Explore the bounded measurable functions with its properties.

CO10: Diagnose Square Summable Function. Create Parseval's identity, Riesz-Fischer theorem.

CO11: Explore Lp-Spaces and the Hölder inequality, Minkowski inequalities.

CO12: Contribute effectively in course-specific interaction.

Approach in teaching:

Interactive Lectures, Discussion, Informative videos

 

Learning activities for the students:

Self learning assignments, Effective questions,  Topic  presentation, Assigned tasks

 

 

Quiz, Class Test, Individual projects,

Open Book Test, Continuous Assessment, Semester End Examination

 

 

 

Unit I: 
Algebras of set:
15.00

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:
15.00

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

 

Unit III: 
Lebesgue integral:
15.00

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:
15.00

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

 

Unit V: 
Lp-spaces and related theorems:
15.00

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.

SUGGESTED READING

  • 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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