Measure Integration and Probability
The goals of the course include:
- understanding the need to introduce a notion of integral more flexible than the Riemann integral
- understanding the construction of the Lebesgue integral
- the ability to apply convergence theorems
- work with random variables and the related integral concepts
Ana Margarida Fernandes Ribeiro
Weekly - 5
Total - 83
M. Capinski, E. Kopp, Measure, Integration and Probability. Springer- Verlag
G. Folland, Real Analysis: Modern Techniques and their Applications, John Wiley & Sons, Second Edition, 1999
J. Lamperti, Probability: A survey of the Mathematical Theory, John Wiley & Sons, Second Edition, 1996
During the theoretical lectures the concepts and results previously introduced in a text for atonomous reading are discussed and developed. Most results are proven. A list of exercises is provided to be solved by the students, being the problem solving classes a place where working groups shall be created.
There are two paper mid-term tests (40%+50%) and 5 online mini-tests (10%). Otherwise the student must pass the final exam. More detailed rules are available in the portuguese version.
Outer measure. Lebesgue measurable sets and Lebesgue measure. Borel sets. Sigma-algebras.
Probabilities: probability space, events, conditional events and independent events.
2- Measurable functions
Lebesgue measurable functions.
Probabilities: random variables. Sigma-algebras generated by random variables. Probability distribution. Independent random variables.
Integral definition. Monotone convergence theorem. Integrable functions. Dominated convergence theorem.
Probabilities: integration with respect to a probability distribution. Absolutely continuous measures. Expectation of a random variable. Characteristic function.
4- Spaces of integrable functions
L^1, L^2, spaces with inner product. Orthogonality and projection. L^p spaces. Complete spaces.
Probabilities: Moments. Independence. Conditional expectation as an orthogonal projection.
5- Product measures
Multidimensional Lebesgue measure. Product sigma-algebras. Product measure. Fubini theorem.
Probabilities: joint distribution. Independence. Conditional probability.
6- Radon-Nykodim theorem
Densities and condicioning. Lebesgue Stieltjes measure. Bounded variation functions. Signed measures.
Probabilities: conditional expectation with respect to a sigma-algebra.
7- Limit theorems
Convergence in probability. Weak law of large numbers. Borel-Cantelli lemma. Strong lawof large numbers. Weak convergence. Central limit theorem.