Measure Theory

Paper Code: 
MAT122
Credits: 
5
Contact Hours: 
75.00
Max. Marks: 
100.00
Objective: 

This course will enable the students to

  1. Understand 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 (COs):

Course

Learning outcomes

(at course level)

Learning and teaching strategies

Assessment

Strategies

Course Code

Course Title

MAT 122

Measure Theory

(Theory)

 

The students will be able to –

 

CO7: Analyse the theory of measure.

CO8: Demostrate Lebesgue integration and its properties.

CO9: Determine Lebesgue theorem on the passage to the limit under the integral sign for bounded measurable functions, Summable functions: Space of square summable functions.

CO10: Know Fourier series and coefficients, Parseval's identity, Riesz-Fisher Theorem, Egoroff's theorem.

CO11: Explain Lp-spaces, Holder - Minkowski inequalities, Completeness of L p -spaces.

CO12: Analyse the concept of Measurable functions: Realization of non-negative measurable function. Structure of measurable functions. Convergence in measure.

 

Approach in teaching:

Interactive Lectures, Discussion, Tutorials, Team teaching

 

Learning activities for the students:

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

Class test, Semester end examinations, Quiz, Presentation

 

 

Unit I: 
I
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: 
II
15.00
Measurable functions: 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: 
III
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: 
IV
15.00
Summable functions: Space of square summable functions. Fourier series and coefficients, Parseval's identity, Riesz-Fisher Theorem.
 
Unit V: 
V
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.
 
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
 
Academic Year: 
2022-2023
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