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 

General characterization





Responsible teacher

Ana Margarida Fernandes Ribeiro


Weekly - 5

Total - 83

Teaching language



The student must have already developed an abstract thought in Analysis and be familiar with Riemann integral, series of functions and countability of sets.


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

Classes consist on theoretical lectures, illustrated by examples and applications, and on problem solving. Most results are proven as well as most of the proposed exercises are solved in the problem solving classes. The remaining are left to the students as part of their learning process.

Evaluation method

The student need to attend 2/3 of the problem solving classes and 2/3 of the lectures to be evaluated.  There are two mid-term tests that can substitute the final exam in case of approval. 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.


Programs where the course is taught: