Measure, Integration and Probability

Basic information

Workload: 

45 hours

Prerequisite: 

Does not exist.

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. 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.

Bibliography

Mandatory: 

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

Complementary: 

· Resnick. The Probability Path. Springer.
· Billingsley. Probability and Measure. Wiley.
· Williams. Probability with Martingales. Cambridge.
· Castro Junior. Measure Theory Course. Euclides Project, IMPA.
· Carlos Isnard. Introduction to Measure Theory. Euclides Project, IMPA.