First to give the concepts of measure, outer measure, measurable set, and measurable function, and then to teach Lebesgue integration of measurable functions.
T. Mısırlıoğlu, Real Analysis Lecture Notes (in Turkish)
R. Bartle, Lebesgue İntegral Kuramına Giriş, Matematik Vakfı Yayınları 5, 1995.
Other Sources
R. G. Bartle, The Elements of Integration and Lebesgue Measure, John Wiley & Sons, Inc., 1966.
G.B. Folland, Real Analysis, Modern Techniques and Their Applications, 2nd Edition, John Wiley & Sons, Inc., 1999.
S. Lang, Real Analysis, 2nd Edition, Addison-Wesley Publihing, 1983.
W. Rudin, Real and Complex Analysis, 3rd Edition, McGraw-Hill, Inc., 1987.
M. R. Spiegel, Theory and Problems of Real Variables, Schaum's Outline Series, McGraw-Hill, Inc., 1990.
E. M. Stein and R. Shakarchi, Real Analysis: Measure Theory, Integration, and Hilbert Spaces, Prentice Lectures in Analysis III, Princeton University Press, 2005.
A. J. Weir, Lebesgue Integration and Measure, Cambridge University Press, 1973.
R.L. Wheeden and A. Zygmund, Measure and Integral: An Introduction to Real Analysis, Marcel Dekker, Inc., 1977.
J. Yeh, Real Analysis: Theory of Measure and Integration, 2nd Edition, World Scientific Publishing, 2006.
Course Schedules
Week
Contents
Learning Methods
1. Week
Ön Bilgiler: Kümeler, fonksiyonlar, doğal sayılar, tam sayılar, bağıntılar, sayılabilir kümeler
Interpreting advanced theoretical and applied knowledge in Mathematics and Computer Science.
PO-2
Critiquing and evaluating data by implementing the acquired knowledge and skills in Mathematics and Computer Science.
PO-3
Recognizing, describing, and analyzing problems in Mathematics and Computer Science; producing solution proposals based on research and evidence.
PO-4
Understanding the operating logic of computer and recognizing computational-based thinking using mathematics as a discipline.
PO-5
Collaborating as a team-member, as well as individually, to produce solutions to problems in Mathematics and Computer Science.
PO-6
Communicating in a foreign language, and interpreting oral and written communicational abilities in Turkish.
PO-7
Using time effectively in inventing solutions by implementing analytical thinking.
PO-8
Understanding professional ethics and responsibilities.
PO-9
Having the ability to behave independently, to take initiative, and to be creative.
PO-10
Understanding the importance of lifelong learning and developing professional skills continuously.
PO-11
Using professional knowledge for the benefit of the society.
Learning Outcomes
LO-1
Reminds the needed preliminaries related to the concepts of sets and
functions, countability, topological properties of the sets in real numbers
and Riemann Integrals for the course of Real Analysis.
LO-2
Understands the concepts of the measure, the null sets, and the outer measure.
LO-3
By understanding Lebesgue measurable sets Lebesgue measure, have a through
knowledge of the properties of Lebesgue measure.
LO-4
Recognize the Borel sets.
LO-5
Analyzing Lebesgue measurable functions, have a through knowledge of the
properties of measurable functions.
LO-6
Learns the concept of Lebesgue integral.
LO-7
Analyzing the concept of integrable functions, proves Monotone and Dominated
Convergence Theorems.
LO-8
Evaluates the relation between Riemann and Lebesgue integrations.
