# Measure Integration and Probability

## Objectives

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

## General characterization

##### Code

7816

##### Credits

6.0

##### Responsible teacher

Ana Margarida Fernandes Ribeiro

##### Hours

Weekly - 5

Total - 83

##### Teaching language

Português

### Prerequisites

Available soon

### Bibliography

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

### Teaching method

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.

### Evaluation method

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.

## Subject matter

1- Measure

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.

3- Integral

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.