Subject: Measure Theory and Probability

Scientific Area:



80 Hours

Number of ECTS:

7,5 ECTS



Overall objectives:

1 - To understand the concepts, results, strategies, income statement and apply the knowledge.
2 - Justify a general framework the knowledge acquired in the course of Probability and Statistics related to the concept of probability.
3 - Acquire the ability to work independently and communicate effectively.
4 - Acquire capacities of reasoning and analysis, rigor, deduction and abstraction.


1 - Motivations and preliminary to the study of measure theory, in particular on the theory of sets. The limitations of the Riemann integral.
2 - Measure
2.1 - Outer dimensions, null sets
2.2 - The Lebesgue measurable sets and Lebesgue measure
2.3 - Lebesgue measure properties
2.4 - Borelian sets
2.5 - Complete measure spaces
2.6 - Probability spaces
2.7 - Discrete and continuous probability, independence of events and sigma-algebras
3 - Measurable functions
3.1 - Measurable functions: definitions and properties
3.2 - Sigma -algebras generated by random variables
3.3 - Probability distributions
4 - Integral
4.1 - Integral: integral properties,
4.2 - Monotone convergence theorem,
4.3 - Dominated convergence theorem.
4.4 - Absolutely continuous measures.
5 - Product measures.
5.1 - Product sigma-algebras.
5.2 - Contruction of the product measure.
5.3 - Fubini's theorem.


Capinski, M. e Kopp , 2000 , Measure, Integral and Probability (Bibliog. Principal) , Springer-verlag
Sandra Mendonça , 2018 , Teoria da Medida e Probabilidade (Notas de apoio às aulas teóricas e lista de exercícios) (Bib. Principal) ,
Adams, M. & Guillemin , 1996 , Measure Theory and Probability. (Bibliog. Secundária) , Birkhäuser..Bauer
Bauer, H. , 2001 , Measure and Integration Theory. (Bibliog. Secundária) , Walter de Gruyter
Burril, C. W. , 1972 , Measure, Integration and Probability. Mc Graw-Hil (Bibliog. Principal) , Mc Graw-Hil
Halmos, P. R. , 1950 , Measure Theory (Bibliog. Secundária) , Van Nostrand
Pollard, D. , 2002 , A User's Guide to Measure Theoretic Probability (Bibliog. Secundária) , Cambridge.Rohatgi,
Rohatgi, V. K. , 1976 , An Introduction to Probability Theory and Mathematical Statistics (Bibliog. Secundária) , John Wiley and Sons

Assesssment methods and criteria:

Classification Type: Quantitativa (0-20)

Evaluation Methodology:
Lectures, problem solving and discussion sessions and presentation of problems. Resolution of exercises. Frequencies: The evaluation aims to measure frequencies and theoretical knowledge and theoretical and practical ability to apply concepts to solve problems. It also aims to analyze the student's ability to relate the subjects studied, whenever feasible. Classes should provide the students interact with the teacher so that they feel free to expose their difficulties and / or contributions to acquire work habits and ability to think independently. The student should be encouraged to maintain a plan of study and work continued throughout the semester. Elements of assessment: 2 Frequencies, each one being worth 50% of the final classification. At the time of appeal the student can improve the classification of a frequency or make a complete examination. In the special time and in note improvement examination the assessment is done through an overall examination.

Subject Leader:

José Luís da Silva