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MEASURE THEORY [1]

Paper Code: 
MAT 122
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

Paper Code

Paper Title

MAT 122

 

 

Measure Theory

(Theory)

 

The students will be able to –

CO6: Students get ideas about the theory of measure.

CO7: Student know how to develop the ideas of Lebesgue integration and its properties

CO8: Students know how to show that certain functions are measurable:

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

CO9: 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. CO10: Students know about Lp-spaces, Holder - Minkowski inequalities, Completeness of Lp-spaces.

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, 1996.
  • Walter Rudin, Real and Complex Analysis, McGraw-Hill Education, 1986.
  • 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 [2], Patrick Fitzpatrick [3], Real Analysis, Pearson’s United States Edition, 2010.
  • G.F. Simmons, Introduction to Topology and Modern Analysis, McGraw-Hill Book, New Delhi, 1963.
  • G. De. Barra, Measure Theory and Integration, Wiley Eastern, 1981.
  • S.K. Berberian, Measure and Integration, McMillan, New York, 1965.
  • I.K. Rana, An Introduction to Measure and Integration, Narosa Publishing House New Delhi, 1997.

 

Academic Year: 
2020-2021 [4]

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Source URL: https://maths.iisuniv.ac.in/courses/subjects/measure-theory

Links:
[1] https://maths.iisuniv.ac.in/courses/subjects/measure-theory
[2] http://www.pearsoned.co.uk/bookshop/Results.asp?iCurPage=1&Type=1&Author=Halsey+Royden&Download=1&SearchTerm=Halsey+Royden
[3] http://www.pearsoned.co.uk/bookshop/Results.asp?iCurPage=1&Type=1&Author=+Patrick+Fitzpatrick&Download=1&SearchTerm=+Patrick+Fitzpatrick
[4] https://maths.iisuniv.ac.in/academic-year/2020-2021