To introduce the concepts of measure and integral with respect to a measure, to show their basic properties, and to provide a basis for further studies in Mathematical Analysis, Probability Measure Theory and Dynamical Systems. To construct Lebesgue's measure and learn the theory of Lebesgue integrals on real line.

**Pre requisite**

Real Analysis

**INTENDED LEARNING OUTCOMES**

Students will be able to

- generalize the idea of length of an interval
- Understand the length of a non-discrete set
- Class of measurable functions
- More general theory of integrals
- Convergence of sequences of measurable functions

**COURSE CONTENTS**

- Lebesgue measure: introduction, outer measure.
- Measurable sets and Lebesgue measure.
- A non-measurable set.
- Measurable functions.
- The Lebesgue integral of a non-negative function.
- The general Lebesgue integral, general measure and integration measure spaces.
- Measurable functions, integration.
- General convergence theorems.

**Recommended Book**

- Royden, H.L. and Fitzpatrick P.M
*, Real Analysis 4th ed.*(NY:Collier Macmillan Co, 2017.)

**Reference Books**

- Philip E.R.
* An introduction to Analysis and Integration Theory. 1st ed.*(USA:

IntextEducationalPublications, 1978.)
- Bartle R.G,
* The Elements of Integration and Lebesgue Measure, 1st Ed.*(Wiley-Interscience. 1995)
- Barra G. De.
* Measure Theory and Integration. 1st ed.(* Ellis, Harwood Ltd, 1981. )

*RESEARCH PROJECT /PRACTICALS/LABS/ASSIGNMENTS*

The projects assigned in this course follow a new approch to theory of integration.

- Problem solving technique for measurable sets.
- Problems related to measurable functions and their properties.
- Different types of convergences of sequence of functions.
- Applications of convergence theorems for Lebesgue integrals.

*ASSESSMENT CRITERIA*

Sessional: 20 (Presentation / Assignment 10, Attendance 05, Quiz 05)

Mid-Term Exam: 30

Final-Term Exam: 50

*Key Dates and Time of Class Meeting*

Wednesday 11:00 am-12:30 pm

Thursday 09:30 am-11:00 am

Commencement of Classes January 13, 2020

Mid Term Examination March 09-13, 2020

Final Term Examination May 04-08, 2020

Declaration of Result May 19, 2020