### Measure Theory (MATH-433 (BS), MATH-627(MSc))

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

1. generalize the idea of length of an interval
2. Understand the length of a non-discrete set
3. Class of measurable functions
4. More general theory of integrals
5. Convergence of sequences of measurable functions

COURSE CONTENTS

1. Lebesgue measure: introduction, outer measure.
2. Measurable sets and Lebesgue measure.
3. A non-measurable set.
4. Measurable functions.
5. The Lebesgue integral of a non-negative function.
6. The general Lebesgue integral, general measure and integration measure spaces.
7. Measurable functions, integration.
8. General convergence theorems.

Recommended Book

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

Reference Books

1. Philip E.R. An introduction to Analysis and Integration Theory. 1st ed.(USA:
IntextEducationalPublications, 1978.)
2. Bartle R.G, The Elements of Integration and Lebesgue Measure, 1st Ed.(Wiley-Interscience. 1995)
3. 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.

1. Problem solving technique for measurable sets.
2. Problems related to measurable functions and their properties.
3. Different types of convergences of sequence of functions.
4. 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