Course Code:
MATH 351
Semester:
Fall
Course Type:
Core
P:
3
Lab:
0
Laboratuvar Saati:
0
Credits:
3
ECTS:
8
Course Language:
English
Course Objectives:
Introduction to measure theory. Learning integration with respect to any measure and Lebesgue Integration and related results.
Course Content:

Measure in plane. Lebesgue measure of sets. Measure as a function of sets. Measurable functions. Egorov and Luzin's theorems. Lebesgue integral. Fatou's theorem. Comparison of Lebesgue and Riemann integrals. Differentiation of indefinite Lebesgue integral. Functions of bounded variations. Radon-Nikodym theorem. Product measures. Fubini's theorem. Lebesgue-Stieltjes and Riemann-Stieltjes integrals Riesz representation theorem.

Course Methodology:
1: Lecture, 2: Problem Solving
Course Evaluation Methods:
A: Written examination, B: Homework

## Vertical Tabs

### Course Learning Outcomes

 Learning Outcomes Teaching Methods Assessment Methods 1) Knows the fundamental theorems in measure theory. 1-2 A-B

### Course Flow

 Week Topics Study Materials 1 Introduction, Extended Real Number System, Measurable Functions and their combinations 2 Measures, Measure Spaces, Charges 3 Simple Functions and Their Integral, Integral of Non-negative Extended Real Valued Measurable Functions 4 Monotone Convergence Theorem, Fatou’s Lemma, Properties of The Integral 5 Integrable Real Valued Functions, Lebesgue Dominated Convergence Theorem 6 Normed Linear Spaces, Lp Spaces, Hölder’s Inequality 7 Minkowski’s Inequality, the Copleteness Theorem 8 Decomposition of Measure 9 Decompositon  of  Measure  Continued 10 Generation of Measures 11 Generation of Measured  Continued 12 Product Measures 13 Product Measures Continued 14 Reviev of the Course

### Recommended Sources

 Textbook The Elements of Integration, Robert G. BARTLE Additional Resources

### Material Sharing

 Documents Assignments Exams

### Assessment

 IN-TERM STUDIES NUMBER PERCENTAGE Mid-terms 2 80 Quizzes 5 10 Assignments 7 10 Total 100 CONTRIBUTION OF FINAL EXAMINATION TO OVERALL GRADE 50 CONTRIBUTION OF IN-TERM STUDIES TO OVERALL GRADE 50 Total 100

 COURSE CATEGORY Expertise/Field Courses

### Course’s Contribution to Program

 No Program Learning Outcomes Contribution 1 2 3 4 5 1 The ability to make computation on the basic topics of mathematics such as limit, derivative, integral, logic, linear algebra and discrete mathematics which provide a basis for the fundamenral research fields in mathematics (i.e., analysis, algebra, differential equations and geometry) x 2 Acquiring fundamental knowledge on fundamental research fields in mathematics x 3 Ability form and interpret the relations between research topics in mathematics x 4 Ability to define, formulate and solve mathematical problems x 5 Consciousness of professional ethics and responsibilty x 6 Ability to communicate actively x 7 Ability of self-development in fields of interest x 8 Ability to learn, choose and use necessary information technologies x 9 Lifelong education x

### ECTS

 Activities Quantity Duration (Hour) Total Workload (Hour) Course Duration (14x Total course hours) 14 3 42 Hours for off-the-classroom study (Pre-study, practice) 14 5 70 Mid-terms (Including self study) 2 15 30 Quizzes 5 1 5 Assignments 7 3 21 Final examination (Including self study) 1 20 20 Total Work Load 188 Total Work Load / 25 (h) 7.52 ECTS Credit of the Course 8