Measure, Integration and Probability

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. Lp spaces. Holder inequality and Minkowski inequality; Completeness of Lp spaces. Average convergence, uniform in almost every point and in Lp. Comparison between the types of convergence. Radon-Nikodym theorem. Product measures and Fubini's theorem. Probability and conditional expectation.

Basic Information

45 hours


Rosenthal. A first look at rigorous probability theory. World Scientific.
Bartle. The Elements of Integration and Lebesgue Measure. Wiley.
Pedro J. Fernandez. Medida e Integração. Coleção Euclides, IMPA.


Resnick. A Probability Path. Springer.
Billingsley. Probability and Measure. Wiley.
Williams. Probability with Martingales. Cambridge.
Castro Junior. Curso de Teoria da Medida. Projeto Euclides, IMPA.
Carlos Isnard. Introdução à Teoria da Medida. Projeto Euclides, IMPA.