Handle measurable sets and become familiar with the extension of a measure (method using the outer measure). Handle integrable functions (dominated convergence theorem).
Pedagogical Objectives: At the end of this course, students should be able to:
1. Know the extension process of a measure.
2. Know the structure of the space of functions.
3. Beppo-Levy, dominated convergence and Fatou theorems. Applications to series of integrable functions.
4. Know the Lebesgue measure on the real line and its relationship with the Riemann integral.
5. The geometric properties of Lp spaces. If possible, Riesz’s theorem.
• More on sets: semi-ring, ring, algebra, σ-algebra, monotone classes, Borel’s σ-algebra.
• Extension of a measure, Carathéodory's theorem, Lebesgue measure.
• Measurable functions: operations (sums, products and single limit). Beppo-Levy, Fatou.
• Image measure.
• Integrable functions, dominated convergence theorem.
• Relation of the Riemann integral to the Lebesgue integral.
• Space Lp (geometric side). Density of K (X) in Lp. Riesz representation theorem.