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

Learning outcomes

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

1. Explain the fundamental concepts of Generalized Measure Theory and Modern Analysis.

   Contribution to Program Outcomes

   1. Having the knowledge about the scope, applications, history, problems, and methodology of mathematics that are useful to humanity both as a scientific and as an intellectual discipline.
   2. Having improved abilities in mathematics communications, problem-solving, and brainstorming skills.
   3. Exhibiting professional and ethical responsibility.

   Method of assessment

   1. Written exam
2. Grasp the Abstract measure space, measurable function and Sigma-additivity.

   Contribution to Program Outcomes

   1. Having the knowledge about the scope, applications, history, problems, and methodology of mathematics that are useful to humanity both as a scientific and as an intellectual discipline.
   2. Having improved abilities in mathematics communications, problem-solving, and brainstorming skills.

   Method of assessment

   1. Written exam
3. Perceive the integral in an abstract measure space and its properties.

   Contribution to Program Outcomes

   1. Having the knowledge about the scope, applications, history, problems, and methodology of mathematics that are useful to humanity both as a scientific and as an intellectual discipline.

   Method of assessment

   1. Written exam
4. Explain the signed measure, Hahn decomposition theorem, The Radon-Nikodym theorem.

   Contribution to Program Outcomes

   1. Having the knowledge about the scope, applications, history, problems, and methodology of mathematics that are useful to humanity both as a scientific and as an intellectual discipline.

   Method of assessment

   1. Written exam

Contents
Week 1:	Abstract measure space, Sigma-additivity
Week 2:	Measurable functions and its properties
Week 3:	Integral in an abstract measure space and its properties
Week 4:	ıntegral in an abstract measure space and its properties
Week 5:	General convergence theorems
Week 6:	Signed measure, Hahn decomposition theorem
Week 7:	Hahn decomposition theorem
Week 8:	Absolutely continuous measure, singular measure
Week 9:	The Radon-Nikodym theorem,Midterm exam
Week 10:	Lebesque decomposition theorem
Week 11:	Lp- spaces
Week 12:	Outer measure, The extension of measure
Week 13:	Inner measure, various convergence types
Week 14:	Fourier series and Fourier integral
Week 15*:	-
Week 16*:	Final exam
Textbooks and materials:	“ Introductory real analysis” A. N. Kolmogorov, S. V. Fomin, translated and edited by Richard A. Silverman.;
Recommended readings:	H.L. Royden, “Real analysis”; “Elements of real analysis” David A. Sprecher.; “Introduction to real analysis” , Robert G. Bartle, Donald R. Sherbert.
	* 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:	9	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:	6	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:	12	2
Mid-term:	2	1
Personal studies for final exam:	12	2
Final exam:	2	1
		Total workload:
		Total ECTS credits:	*
	* ECTS credit is calculated by dividing total workload by 25. (1 ECTS = 25 work hours)