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Description of Individual Course Units

Course Unit Code Course Unit Title Type Of Course D U L ECTS MAT 4046 MEASURE THEORY AND LEBESQUE INTEGRAL ELECTIVE 4 0 0 7

Offered By

Mathematics

Level of Course Unit

First Cycle Programmes (Bachelor's Degree)

Course Coordinator

ASSISTANT PROFESSOR SEÇIL GERGÜN

Offered to

Mathematics (Evening) Mathematics

Course Objective

The aim of the course is to introduce the basic concepts of Measure theory and Lebesgue integral.

Learning Outcomes of the Course Unit

1   to define measure on a sigma algebra. 2   to distinguish Lebesgue measurable sets. 3   to distinguish measurable functions. 4   to relate modes of convergence: uniform, almost everywhere. 5   to define Lebesgue integral. 6   to relate Lebesgue and Riemann integral. 7   to define L_p spaces.

Mode of Delivery

Face -to- Face

Prerequisites and Co-requisites

None

Recomended Optional Programme Components

None

Course Contents

Week Subject Description 1 Review for Riemann integrability. Reasons for the development of Lebesgue integral. 2 Systems of sets. General measure theory. 3 Measure on the real line. Lebesgue measure. 4 Outer and inner measure. Examples of measurable sets. 5 Approximation of measurable sets. Non-measurable, non-Borel sets. 6 The properties of measurable functions. 7 Simple function. Equivalent functions. 8 Modes of convergence. Egorov's Theorem. 9 Lebesgue integral of simple functions. 10 Lebesgue integral of measurable functions. 11 Passage to the limit in Lebesgue integral. 12 Lebesgue-Stieljies integral, Riemann-Stieljies integral. 13 Definition and basic properties of the spaces L_p and L_2. 14 Riesz Represantation Theorem.

Recomended or Required Reading

Textbook(s): The Elements of Integration and Lebesgue Measure, Robert G. Bartle, John Wiley & Sons, 1995. Supplementary Book(s): Real Analysis, 3rd Edition; H.L. Royden, Macmillan Publishing Company, 1988. References: Introductory Real Analysis, A. N. Kolmogorov, S.V. Fomin, Dover Publications, 1970.

Planned Learning Activities and Teaching Methods

Lecture Notes Text Book(s) Solving Problems

Assessment Methods

SORTING NUMBER SHORT CODE LONG CODE FORMULA 1 MTE MIDTERM EXAM 2 QUZ QUIZ 3 FIN FINAL EXAM 4 FCGR FINAL COURSE GRADE (RESIT) MTE * 0.30 + ASG * 0.30 + FIN * 0.40 5 RST RESIT 6 FCGR FINAL COURSE GRADE (RESIT) MTE * 0.30 + ASG * 0.30 + RST * 0.40 *** Resit Exam is Not Administered in Institutions Where Resit is not Applicable.

Further Notes About Assessment Methods

None

Assessment Criteria

To be announced.

Language of Instruction

English

Course Policies and Rules

To be announced.

Contact Details for the Lecturer(s)

e-mail: sedef.erim@deu.edu.tr, tel: (232) 301 85 91

Office Hours

To be announced.

Work Placement(s)

None

Workload Calculation

Activities Number Time (hours) Total Work Load (hours) Lectures 14 4 56 Preparations before/after weekly lectures 14 4 56 Preparation for midterm exam 1 25 25 Preparation for final exam 1 30 30 Final 1 2 2 Midterm 1 2 2 Quiz etc. 4 1 4 TOTAL WORKLOAD (hours) 175

Contribution of Learning Outcomes to Programme Outcomes

PO/LO PO.1 PO.2 PO.3 PO.4 PO.5 PO.6 PO.7 PO.8 PO.9 PO.10 PO.11 PO.12 PO.13 LO.1 5 5 4 4 5 3 4 LO.2 5 5 4 4 5 3 4 LO.3 5 5 4 4 5 3 4 LO.4 5 5 4 4 5 3 4 LO.5 5 5 4 4 5 3 4 LO.6 5 5 4 4 5 3 4 LO.7 5 5 4 4 5 3 4