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



Available soon


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 authonomous 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 to discussion.

Evaluation method

There are two paper mid-term tests (50%+50%). 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: