Iğdır University Lifelong Learning Programme

Description of Individual Course Units

Course Unit Code	Course Unit Title	Type of Course Unit	Year of Study	Semester	Number of ECTS Credits
MAT-23-126	REAL ANALYSIS I	Elective	1	1	6

Level of Course Unit

Second Cycle

Objectives of the Course

The purpose of this course is to explain the measure in the general sense and the concept of the measurable sets and functions, the class of equivalentness of functions. Properties of the measurable sets and functions, moreover properties of the spaces of the measurable sets and functions.

Name of Lecturer(s)

Prof. Dr. Kamal SOLTANOV

Learning Outcomes

1	To teach the students the general concept of the measure and to show why this generalization is needed
2	To teach the students the properties of the measurable sets and functions
3	To teach the students the properties of the spaces of the measurable sets and functions and their relations

Mode of Delivery

Daytime Class

Prerequisites and co-requisities

It is need to know the basic facts of Mathematical Analysis, Linear Algebra, Real analysis, Algebra and Topology in the undergraduate level.

Recommended Optional Programme Components

Course Contents

The basic properties of the set theory About the metric space Topology and continuity General approach to the measure on R1, algebra of Borel, the general measure on R1 and its properties General approach to the measure on Rn and its properties The ring (σ-ring) and algebra (σ-algebra) of the subsets of the general sets, the measure on the set of the subsets, the σ-additive measure and its properties The outer measure of Lebesgue, the Carateodory approach to the Lebesgue measure and properties of this measure The set of the Lebesgue measurable sets is metric space The class of the measurable functions in the Lebesgue sense and their structure Properties of the set of the measurable functions (the algebraical operations in this set) The convergence in the set of the measurable functions and the metric on this set Egorov`s theorem and convergence of the µ-uniformly Luzin`s theorem and preparation to the final exam

Weekly Detailed Course Contents

Week	Theoretical	Practice	Laboratory
1	The basic properties of the set theory
2	About the metric space
3	Topology and continuity
4	General approach to the measure on R1, algebra of Borel, the general measure on R1 and its properties
5	General approach to the measure on Rn and its properties
6	The ring (σ-ring) and algebra (σ-algebra) of the subsets of the general sets, the measure on the set of the subsets, the σ-additive measure and its properties
7	MID TERM EXAM
8	The outer measure of Lebesgue, the Carateodory approach to the Lebesgue measure and properties of this measure
9	The set of the Lebesgue measurable sets is metric space
10	The class of the measurable functions in the Lebesgue sense and their structure
11	Properties of the set of the measurable functions (the algebraical operations in this set)
12	The convergence in the set of the measurable functions and the metric on this set
13	Egorov`s theorem and convergence of the µ-uniformly
14	Luzin`s theorem and preparation to the final exam
15	Final Exam

Recommended or Required Reading

Natanson I. P - Theory of Function of Real Variable. New York , 1959-1961 Royden, H. L. - Real Analysis. Mac Millan New York 1968. Kolmogorov A. N, Fomin S. V. - Introductory Real Analysis. New York ,1975 Rao M. M. - Measure Theory and İntegration. New York Wiley, 1984 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

Planned Learning Activities and Teaching Methods

Assessment Methods and Criteria

Term (or Year) Learning Activities	Quantity	Weight
Midterm Examination	1	100
SUM		100
End Of Term (or Year) Learning Activities	Quantity	Weight
Final Examination	1	100
SUM		100
Term (or Year) Learning Activities		50
End Of Term (or Year) Learning Activities		50
SUM		100

Language of Instruction

Turkish

Work Placement(s)

Workload Calculation

Activities	Number	Time (hours)	Total Work Load (hours)
Midterm Examination	1	1	1
Final Examination	1	2	2
Attending Lectures	14	3	42
Question-Answer	14	3	42
Individual Study for Mid term Examination	1	50	50
Individual Study for Final Examination	1	50	50
TOTAL WORKLOAD (hours)			187

Contribution of Learning Outcomes to Programme Outcomes

	PO 1	PO 2	PO 3	PO 4	PO 5	PO 6	PO 7	PO 8	PO 9	PO 10	PO 11
LO1	5	5	5	5	5	5	5	5	5	5	5
LO2	5	5	5	5	5	5	5	5	5	5	5
LO3	5	5	5	5	5	5	5	5	5	5	5

* Contribution Level : 1 Very low 2 Low 3 Medium 4 High 5 Very High