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

Maria Fernanda de Almeida Cipriano Salvador Marques


Weekly - 4

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


Attendance of 2/3 of the classes is mandatory for approval.


 Continuous evaluation

During the semester two tests will be carried out. Each test is rated up to a maximum of 20 values.

1st Test: all students enrolled in the course may present themselves to the 1st test.

2nd Test: all students enrolled in the discipline who have obtained frequency or are exempt from it may appear at the 2nd test.

The classification of the Continuous Evaluation is obtained by making the arithmetic mean of the classifications obtained in the 2 tests.

If the Continuous evaluation classification is greater than, or equal to, 9.5 the student is approved with that classification rounded up to the units. 

If the classification of the Continuous Assessment is less than 9.5, the student may take the Exam.



All students enrolled in the discipline who have obtained Frequency or are exempted from it, and who have not passed the continuous assessment, may take the Exam. 

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.