Analysis 3- Measure Theory

Abstract

Abstract measure and integration theory, including: Carathéodory’s theorem, Lebesgue measure, Radon measures, Hausdorff measure, convergence theorems, L^p spaces, product measures, and Fubini’s theorem.

Objective

1 Explain the motivation for measure theory and how it extends classical notions of length, area, and integration. 2 Define σ-algebras, measurable sets, measures, and measurable functions, and give concrete examples. 3 Construct the Lebesgue measure on R^n and compute simple examples. 4 Develop the Lebesgue integral for simple and general functions. 5 Apply the key convergence theorems: Monotone Convergence, Fatou’s Lemma, Dominated Convergence, and Vitali Convergence Theorem. 6 Understand the relationship between Lebesgue and Riemann integration. 7 Use product measures and apply Fubini’s Theorem to compute double integrals. 8 Explore L^p spaces and their properties, including inequalities and completeness. 9 Write clear, rigorous proofs and explanations involving measurable sets, functions, and integrals.

Content

• Measure Spaces (Lebesgue Measure, Hausdorff Measure, Radon Measure) • Measurable Functions: definition and properties • Integration: definition, properties, theorems of convergence, Lebesgue L^p spaces • Product Measures and Multiple Integrals. Fubini and Tonelli Theorems, Convolutions • Differentiation of measures (if time permits)

Resources