Advanced Real Analysis-I

Paper Code: 
MAT324 C
Credits: 
5
Contact Hours: 
75.00
Max. Marks: 
100.00
Objective: 

This course will enable the students to –

  1. Explore their knowledge in the area of real analysis. 
  2. Get sufficient knowledge of the subject which can be used by students for further applications in their respective domains of interest.
  3. Understand the Introduction of Ordinal number, perfect sets, Borel measureable functions.
  4. Get ideas about approximate continuous function, Henstock integration.

Course Outcomes (COs):

Course

Learning outcomes

(at course level)

Learning and teaching strategies

Assessment

Strategies

Course Code

Course Title

 

 

 

MAT 324C

 

 

 

Advanced Real Analysis-I

   (Theory)

 

 

 

 

The students will be able to –

 

CO111: Introduce the concept of ordinal numbers.

CO112: Describe properties of perfect set and prove related theorems.

CO113: Discuss properties of  Borel measurable functions and Darbous function of Baire class one

CO114: Analyze characteristics of approximate continuous function.

CO115: Understand the concept of Henstock integration on the real line.

Approach in teaching:

Interactive Lectures, Discussion, Power Point Presentations, Informative videos

Learning activities for the students:

Self learning assignments, Effective questions, presentations, Field trips

Quiz, Poster Presentations,

Power Point Presentations, Individual and group projects,

Open Book Test, Semester End Examination

 

Unit I: 
I
15.00
Ordinary number: Order types, Well-ordered sets, Transfinite induction, Ordinal numbers, Comparability of ordinal numbers, Arithmetic of ordinal numbers, First uncountable ordinal Ω.
 
Unit II: 
II
15.00
Descriptive properties of sets: Perfect sets, Decomposition of a closed set in terms of perfect sets of first category, 2nd category and residual sets, Characterization of a residual set in a compete metric space, Borel sets of class α, ordinal α < Ω,  Density point of a set in R, Lebesgue density theorem.
 
Unit III: 
III
15.00
Functions of some special classes: Borel measurable functions of class α (α < Ω) and its basic properties, Comparison of Baire and Borel functions, Darboux functions of Baire class one.
 
Unit IV: 
IV
15.00
Continuity: Nature of the sets of points of discontinuity of Baire one functions, Approximate continuity and its fundamental properties, Characterization of approximate continuous functions.
 
Unit V: 
V
15.00
Henstock integration on the real line:Concepts of δ-fine partition of the closed interval [a,b] where δ is a positive function on [a,b], Cousin’s lemma, definition of Henstock integral of a functions over the interval [a,b] and its basic properties, Saks-Henstock lemmas and its applications, Continuity of the indefinite integral, Fundamental theorem, Convergence theorems, Absolute Henstock integrability, Characterization of Lebesgue integral by absolute Henstock integral.
 
Essential Readings: 
  • A.M. Bruckner, J.B. Bruckner and B.S. Thomson, Real Analysis, Prentice-Hall, New York 1997.
  • H.S. Gaskill and P.P. Narayanswami, Elements of Real Analysis, PHI, 1988.
  • W.P. Parzynski and P.W. Zipse, Introduction to Mathematical Analysis, MC Graw-Hill Company, 1982.
  • I.P. Natanson, Theory of Functions and Real Variable, Vol. I& II, Frederic Ungar Publishing, 1955.
  • C. Goffman, Real Functions, Rinehart Company, N.Y. 1953
References: 
  • P.Y. Lee, Lanzhou Lectures on Henstock Integration, World Scintific Press, 1990.
  • J.F. Randolph, Basic Real and Abstract Analysis, Academic Press, N.Y. 2014.
  • S.M. Srivastava, A Course on Borel Sets, Springer,N.Y. 1998.
  • R.G. Rartle, Introduction to Real Analysis, John Willey and Sons, 2000.
  • A.J. Kosmala, Introductory Mathematical Analysis, WCB Company, 1995.
 
Academic Year: 
2022-2023
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