Real Function Theory

February, 2006 to July, 2006

Teaching Progress

Chapter 1 Preliminaries

Lecture 1. Cardinality of Set

Lecture 2. Real numbers

Lecture 3. Euclidean Spaces

Lecture 4. Euclidean Spaces (concrete discussion)

Chapter 2. Measure in Euclidean Space

Lecture 1. Lebesgue Outer Measure

Lecture 2. Lebesgue Measure

Lecture 3. Measurable Sets

Lecture 4. Measure and Topology

Lecture 5. A Nonmeasurable Set

Chapter 3. Lebesgue Measurable Functions   

Lecture 1. Definition and elementary properties

Lecture 2. Semicontinuous functions

Lecture 3. Egorov's theorem and Lusin's theorem

Lecture 4. Convergence in measure

Chapter 4 Lebesgue Integral

Lecture 1. Definition and elementary properties

Lecture 2. Integrable functions 

Lecture 3. Riemann Integral

Lecture 4. Repeated integration

Lecture 5. Application of Fubini's Theorem

Lecture 6. Supplement--- On the continuous extension of functions in Rⁿ

Chapter 5. Function classes and function spaces

Lecture 1. Functions of Bounded Variation

Lecture 2. Monotone Functions

Lecture 3. Hardy-Littlewood maximal functions

Lecture 4. Absolutely continuous functions

Lecture 5. Spaces Lp(Rn); 1≤p≤∞

Lecture 6. Smoothness of functions in Lp(Rn)

Lecture 7. Convolution, Identity Approximation

Students who present reports in the Seminars