MEASURE THEORY

Paper Code: 
MAT122
Credits: 
5
Contact Hours: 
75.00
Max. Marks: 
100.00
Objective: 
This course will enable the students to -
  1. Gain understanding of the abstract measure theory and definition and main properties of the integral. 
  2. To construct Lebesgue's measure on the real line and in n-dimensional Euclidean space.
  3. To explain the basic advanced directions of the theory.

Learning Outcomes

Learning and teaching strategies

Assessment

After the completion of the course the students will be able to:

CLO6- students get ideas about the theory of measure.

CLO7- Student know how to develop the ideas of Lebesgue integration and its properties                   

CLO8- students know how to show that certain functions are measurable:

  • construct the Lebesgue integral
  • understand properties of the Lebesgue integral

CLO9- Students know about Lebesgue theorem on the passage to the limit under the integral sign for bounded measurable functions, Summable functions: Space of square summable functions. Fourier series and coefficients, Parseval's identity, Riesz-Fisher Theorem.

CLO10- Students know about Lp-spaces, Holder - Minkowski inequalities, Completeness of L p -spaces.

Approach in teaching:

Interactive Lectures, Discussion, Tutorials, Reading assignments, Demonstration, Team teaching

Learning activities for the students:

Self learning assignments, Effective questions, Simulation, Seminar presentation, Giving tasks, Field practical

 

 

 

 

 

 

 

Presentations by Individual Students.

Class Tests at Periodic Intervals.

Written assignment(s)

Semester End Examination

 

 

 

 

 

 

 

 

 

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, 1996.
  • Walter Rudin, Real and ComplexAnalysis, McGraw-Hill Education, 1986.
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, NewDelhi, 1963.
  • G.De.Barra, Measure Theory and Integration, Wiley Eastern, 1981.
  • S.K.Berberian, Measure and Integration, McMillan, NewYork,1965.
  • I.K.Rana,An Introduction to Measure and Integration, Narosa Publishing House New Delhi,1997.
Academic Year: 
2019-2020
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