MAT 332E - Real Analysis I

Course Objectives

To introduce the measure of plane sets first, and then the measure in general. To introduce the Riemann integral and the Lebesgue integral as an extension. To teach properties of Lebesgue integral and give ability to apply on mathematical problems

Course Description

1. Intervals on the line. Length.
2. Algebras, $\sigma$-algebras. Notion of measure.
3. Outer and inner measures.
4. The extension of measure function. General measure theory.
5. Properties of measurable sets. Cantor and Borel sets.
6. Measurable functions. Simple functions.
7. Lebesgue’s integral. Properties of Lebesgue’s integral.
8. Comparison of Lebesgue’s with Riemann’s integral.
9. Notions of convergence. Lebesgue’s Theorem.
10. Fatou's Lemma and Levi's Theorems.
11. Fubini’s Theorem.