

Contents


Week 1: 
Preliminary. 
Week 2: 
The concept of measure. 
Week 3: 
Lebesgue outer measure. 
Week 4: 
Measurable and nonmeasurable set. 
Week 5: 
Measurable function and its properties. 
Week 6: 
MIDTERM 1. 
Week 7: 
Egorov's and Lusin's theorems. 
Week 8: 
The convergence of sequence of measurable functions under various kinds of convergence. 
Week 9: 
Lebesgue integration for simple functions. 
Week 10: 
General Lebesgue integral, Lebesgue's, Levi’s and Fatou’s theorems. 
Week 11: 
Relation between the Riemann and Legesgue integrals. 
Week 12: 
MIDTERM 2. 
Week 13: 
Lebesque spaces and their properties, the relation between different Lp spaces. 
Week 14: 
Dual spaces of Lebesgue spaces. 
Week 15*: 
General review. 
Week 16*: 
Final exam. 
Textbooks and materials: 
• Kolmogorov A.N., Fomin S.V., Introductory Real Analysis, PrenticeHall, 1970. • Riesz and B. Sz.Nagy, Functional Analysis, Dover Publications, 1990. • Royden, H. L., Real Analysis. Mac Millan New York 1968. • Lusternik L.A., Sobolev V.J., Elements of Functional Analysis, John Wiley & Sons, 1974 F. • Natanson I. P., Theory of Function of Real Variable. New York,1960. • Rao M. M., Measure Theory and İntegration, John Wiley, New York, 1987. • Shilov G. E., Gurevich B. L., Integral, Mesure and Derivative: A unified approach. PrenticeHall, 1966. • Howes N. R., Modern Analysis and Topology. SpringerVerlag, 1995.

Recommended readings: 
• Kolmogorov A.N., Fomin S.V., Introductory Real Analysis, PrenticeHall, 1970. • Riesz and B. Sz.Nagy, Functional Analysis, Dover Publications, 1990. • Royden, H. L., Real Analysis. Mac Millan New York 1968. • Lusternik L.A., Sobolev V.J., Elements of Functional Analysis, John Wiley & Sons, 1974 F. • Natanson I. P., Theory of Function of Real Variable. New York,1960. • Rao M. M., Measure Theory and İntegration, John Wiley, New York, 1987. • Shilov G. E., Gurevich B. L., Integral, Mesure and Derivative: A unified approach. PrenticeHall, 1966. • Howes N. R., Modern Analysis and Topology. SpringerVerlag, 1995.


* Between 15th and 16th weeks is there a free week for students to prepare for final exam.

