Yozgat Bozok 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
MAT565	REEL ANALİZ I	Elective	1	1	6

Level of Course Unit

Second Cycle

Objectives of the Course

To introduce to the student concept of Lebesgue integral, and applications.

Name of Lecturer(s)

Doç. Dr. Abdullah SÖNMEZOĞLU

Learning Outcomes

1	To prove theorems about measurable sets and measurable functions
2	Identify and apply the concept of Lebesgue integral
3	To learn the relationship between Lebesgue integral and Riemann integral

Mode of Delivery

Formal Education

Prerequisites and co-requisities

None

Recommended Optional Programme Components

None

Course Contents

Metric spaces, Open and closed sets, Stacking points of a set, discrete points, closure and interior, Sequences and convergence, Stacking and limit points of sequences, Real number sequences, liminf and limsup, Bolzano-Weierstrass theorem, Cauchy sequences and completeness, Metric spaces compact, compact characterization, sequential characterization of compact sets, bounded and fully bounded sets, Heine-Borel theorem in R, Continuous functions, Image and inverse images of open, closed, compact sets under continuous functions, characterization of continuous functions, uniform continuity and Cauchy sequences, Point and uniform convergence of function sequences, Continuous function sequences, Spaces of continuous functions defined on a uniform convergent and compact K metric space C (K), Conjugality, compactness and Ascoli-Arzela theorem, Density and Stone-Weierstrass theorem, Derivative , Vitali cover lemma and monotonous functions, Limited oscillation functions, Absolute continuous functions, Lipschitz functions, Riemann integral, Step functions and Riemann sums, Uniform convergent function sequences and integrals, Point convergence and Egoroff theorem, Limited convergence theorem

Weekly Detailed Course Contents

Week	Theoretical	Practice	Laboratory
1	Sigma Algebra and Algebra Concepts
2	Axiom of Choice and the Infinite Direct Products
3	Countable sets
4	Real Number System
5	Open and Closed Sets
6	Borel Sets
7	Measurement and exterior Measurement Concepts
8	Midterm Exam
9	Measurement and exterior Measurement Concepts
10	Measurement and exterior Measurement Concepts
11	Measurement and exterior Measurement Concepts
12	Measurable sets and Lebesgue Measure
13	Measurable sets and Lebesgue Measure
14	Measurable sets and Lebesgue Measure
15	Clusters can not be measured
16	Final Exam

Recommended or Required Reading

H. L. Royden, Real Analysis, Macmillan Publishing Co. Inc., 1963. A. Mukherjea and K. Pothoven, Real and Functional Analysis, Plenum Press, 1984. M. Balcı, Real Analysis, Balcı Yayınları, 2000. A. Dönmez, Real Analysis, Seçkin Publishing, 2001.

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		40
End Of Term (or Year) Learning Activities		60
SUM		100

Language of Instruction

Work Placement(s)

None

Workload Calculation

Activities	Number	Time (hours)	Total Work Load (hours)
Midterm Examination	1	2	2
Final Examination	1	2	2
Self Study	14	5	70
Individual Study for Mid term Examination	7	8	56
Individual Study for Final Examination	7	8	56
TOTAL WORKLOAD (hours)			186

Contribution of Learning Outcomes to Programme Outcomes

LO1

LO2

LO3

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