Introduction the Measure Theory

Syllabus: 

Measurable sets and measurement spaces. Construction of measures. Measurement extensions, Caratheodory Theorem. Lebesgue and Lebesgue-Stieltjes measure. Measurable Functions. Simple functions. Lebesgue integral. Monotonous convergence theorem, Fatou's lemma and dominated convergence theorem. Comparison between Riemann and Lebesgue integrals. Product measures and Fubini's theorem. Radon-Nikodym theorem. Lp spaces. Holder inequality and Minkowiski inequality; Completeness of Lp spaces. Convergence on average, uniform in almost every point and in Lp. Comparison between the types of convergence.

Bibliography

Mandatory: 

•    Measure theory course, Castro Junior, A. Armando .; Projeto Euclides Collection, 2008; 
•    Introduction to measurement and integration, Isnard Carlos; Projeto Euclides Collection, 2007; 
•    Real and Complex Analysis, Rudin, W .; McGraw-Hill 1986. 

Complementary: 

•    Measure Theory and Integration, Michael E. Taylor, American Mathematical Society, 2006; 
•    Measure Theory: A First Course, Carlos S Kubrusly, Academic Press, 2007; 
•    Real analysis and probability, Dudley, Cambridge studies in advanced Math, R. M, 2000; 
•    Real Analysis, Royden, H.L., New York: Addison Wesley, 1988;
•    The Elements of Integration and Lebesgue Measure, Robert G. Bartle, Wiley, 1995.