• Basic information
• Learning outcomes
• Contents
• Assessment
• Workload

Learning outcomes

Upon successful completion of this course, students will be able to:

1. Explain the concept of measure, the measurable set, measurable function and know how to apply.

   Contribution to Program Outcomes

   1. Define and manipulate advanced concepts of Mathematics
   2. Relate mathematics to other disciplines and develop mathematical models for multidisciplinary problems
   3. Acquire scientific knowledge and work independently
   4. Develop mathematical, communicative, problem-solving, brainstorming skills.
   5. Effectively express his/her research ideas and findings both orally and in writing
   6. Demonstrating professional and ethical responsibility.

   Method of assessment

   1. Written exam
2. Analyze the convergence of sequence of measurable functions under various kinds of convergence.

   Contribution to Program Outcomes

   1. Define and manipulate advanced concepts of Mathematics
   2. Relate mathematics to other disciplines and develop mathematical models for multidisciplinary problems
   3. Acquire scientific knowledge and work independently
   4. Develop mathematical, communicative, problem-solving, brainstorming skills.
   5. Effectively express his/her research ideas and findings both orally and in writing
   6. Demonstrating professional and ethical responsibility.

   Method of assessment

   1. Written exam
3. Explain the properties of Lebesque integral and relation between Lebesque spaces

   Contribution to Program Outcomes

   1. Define and manipulate advanced concepts of Mathematics
   2. Relate mathematics to other disciplines and develop mathematical models for multidisciplinary problems
   3. Acquire scientific knowledge and work independently
   4. Develop mathematical, communicative, problem-solving, brainstorming skills.
   5. Effectively express his/her research ideas and findings both orally and in writing
   6. Demonstrating professional and ethical responsibility.

   Method of assessment

   1. Written exam

Contents
Week 1:	Preliminary.
Week 2:	The concept of measure.
Week 3:	Lebesgue outer measure.
Week 4:	Measurable and nonmeasurable set.
Week 5:	Measurable functions.
Week 6:	Properties of measurable functions. Midterm Exam 1.
Week 7:	Egorov's and Lusin's theorems.
Week 8:	The convergence of sequence of measurable functions under various kinds of convergence.
Week 9:	Lebesgue integration for simple functions.
Week 10:	General Lebesgue integral.
Week 11:	Lebesgue's, Levi’s and Fatou’s theorems.
Week 12:	Relation between the Riemann and Legesgue integrals. Midterm Exam 2.
Week 13:	Lebesque spaces and their properties, the relation between different Lp spaces.
Week 14:	Dual spaces of Lebesgue spaces.
Week 15*:	-
Week 16*:	Final Exam.
Textbooks and materials:	• Kolmogorov A.N., Fomin S.V., Introductory Real Analysis, Prentice-Hall, 1970. • Riesz and B. Sz.-Nagy, Functional Analysis, Dover Publications, 1990. • Royden, H. L., Real Analysis. Mac Millan New York 1968. • Lusternik L.A., Sobolev V.J., Elements of Functional Analysis, John Wiley & Sons, 1974 F. • Natanson I. P., Theory of Function of Real Variable. New York,1960. • Rao M. M., Measure Theory and İntegration, John Wiley, New York, 1987. • Shilov G. E., Gurevich B. L., Integral, Mesure and Derivative: A unified approach. Prentice-Hall, 1966. • Howes N. R., Modern Analysis and Topology. Springer-Verlag, 1995.
Recommended readings:	• Kolmogorov A.N., Fomin S.V., Introductory Real Analysis, Prentice-Hall, 1970. • Riesz and B. Sz.-Nagy, Functional Analysis, Dover Publications, 1990. • Royden, H. L., Real Analysis. Mac Millan New York 1968. • Lusternik L.A., Sobolev V.J., Elements of Functional Analysis, John Wiley & Sons, 1974 F. • Natanson I. P., Theory of Function of Real Variable. New York,1960. • Rao M. M., Measure Theory and İntegration, John Wiley, New York, 1987. • Shilov G. E., Gurevich B. L., Integral, Mesure and Derivative: A unified approach. Prentice-Hall, 1966. • Howes N. R., Modern Analysis and Topology. Springer-Verlag, 1995.
	* Between 15th and 16th weeks is there a free week for students to prepare for final exam.

Assessment
Method of assessment	Week number	Weight (%)
Mid-terms:	6,12	40
Other in-term studies:		0
Project:		0
Homework:		0
Quiz:		0
Final exam:	16	60
	Total weight:	(%)

Workload
Activity	Duration (Hours per week)	Total number of weeks	Total hours in term
Courses (Face-to-face teaching):	3	14
Own studies outside class:	7	14
Practice, Recitation:	0	0
Homework:	0	0
Term project:	0	0
Term project presentation:	0	0
Quiz:	0	0
Own study for mid-term exam:	10	2
Mid-term:	6	2
Personal studies for final exam:	18	1
Final exam:	3	1
		Total workload:
		Total ECTS credits:	*
	* ECTS credit is calculated by dividing total workload by 25. (1 ECTS = 25 work hours)