Examination procedure
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Examination procedure notes
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Prerequisites
The course is targeted to Math students in the III year, but also Physics students are welcome
Syllabus
ABSTRACT MEASURE THEORY
Rings, algebras, and sigma-algebras
Additive and sigma-additive set functions
Measurable and measure spaces
Measure constructions and the Dynkin Lemma
Hausdorff and Lebesgue measures
Regularity properties
INTEGRATION
Simple and measurable functions
Integral of measurable functions
Lusin, Egorov, and Ulam theorems
Jensen's inequality
Convergence of integrals: Levi, Fatou, Lebesgue, and Vitali-Hahn-Saks
MEASURE OPERATIONS
Transport and change-of-variables formula
Products: Fubini-Tonelli theorem
Radon-Nikodym theorem
Covering theorems and applications to the differentiation of measures
Fundamental theorem of integral calculus
FUNCTIONAL ANALYSIS
Hilbert spaces: Riesz theorem, Bessel's inequality, Parseval's identity
Banach spaces:
- Banach-Steinhaus, open mapping, and closed graph theorems
- reflexivity, weak and strong topologies, Banach-Alaoglu-Bourbaki theorem, Mazur's lemma.
DUALITY AND WEAK CONVERGENCE
L^p spaces: compactness criteria with respect to strong and weak convergence
Dual of C(K) and applications, weak convergence criteria of measures, and Prokhorov's theorem
Bibliographical references
H. Brezis: Analyse Fonctionelle
W. Rudin: Real and Complex Analysis
L.Ambrosio, G.Da Prato, A.Mennucci: Introduction to Measure Theory and Integration