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 | 3 | 6 |
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Level of Course Unit |
Third 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 |
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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 |
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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
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Weekly Detailed Course Contents |
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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 | | |
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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
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Planned Learning Activities and Teaching Methods |
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Assessment Methods and Criteria | |
Midterm Examination | 1 | 100 | SUM | 100 | |
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) | |
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Workload Calculation |
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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 |
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Contribution of Learning Outcomes to Programme Outcomes |
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 |
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* Contribution Level : 1 Very low 2 Low 3 Medium 4 High 5 Very High |
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Iğdır University, Iğdır / TURKEY • Tel (pbx): +90 476
226 13 14 • e-mail: info@igdir.edu.tr
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