Prerequisites: knowledge of (undergraduate) real analysis

Description

• Outer measure, measure, -algebras, Dynkin Theorem, Carathéodory's Extension Theorem
• Borel measures, Lebesgue measure
• Measurable functions, Lebesgue integral (Monotone Convergence Theorem, Fatou's Lemma, Dominated Convergence Theorem)
• Modes of Convergence (Egoroff's Theorem, Lusin's Theorem)
• Product Measures (Fubini-Tonelli Theorems), n-dimensional Lebesgue integral
• Signed Measures (Hahn Decomposition, Jordan Decomposition, Radon-Nikodym Theorem, change of variables)
• Differentiation (Lebesgue Differentiation Theorem)
• Spaces (Hölder's inequality, Minkowskii's inequality, completeness, equiintegrability (uniform integrability), Vitali's convergence theorem)