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Be English-friendly or any other language-friendly means that UC is taught in a language but can either of the
following conditions:
1. There are support materials in English / other language;
2. There are exercises, tests and exams in English / other language;
3. There is a possibility to present written or oral work in English / other language.
1
4.0
10.0 h/sem
6.0 h/sem
0.0 h/sem
0.0 h/sem
0.0 h/sem
0.0 h/sem
0.0 h/sem
16.0 h/sem
96.0 h/sem
0.0 h/sem
112.0 h/sem
Since year
2019/2020
Pre-requisites
No
Objectives
At a first level, provide an understanding of the concepts, terminology and meaning of the main theorems in this area, so as to make the masters self-sufficient in later studies; On a second level, provide some training in reasoning and calculations more relevant in this area of mathematics in order to provide the graduate student's ability to rigorous justification of its conclusions; On a third level, more ambitious, awakening the ability to conceive proofs in problem solving.
Program
1. Sigma-algebras. Mesurable spaces and functions. 2. Finite and sigma-finite measures. Measure properties. Measure and probability spaces. 3. Integral of a function on a measure space. Integral properties. Integrability. 4. Lebesgue integral on the real line. 5. Comparison with Riemann integral. 6. Product measures and Fubini's theorem. 7. Density functions and associated measures. 8. Theorem of Radon-Nikodym. 9. Change of variables. The spaces L1 and L2. 10. Sequence of functions convergence.
Evaluation Method
Regular grading system: - One individual exam (100%) It is considered that a student has course approval if he has a grade equal or above 9.5 points.
Teaching Method
The student should acquire analytical, information gathering, written and oral communication skills, through the following learning methodologies (LM): 1. Expositional, to the presentation of the theoretical reference frames 2. Participative, with analysis and resolution of application exercises 3. Self-study, related with autonomous work by the student, as is contemplated in the Class Planning.
Observations
Basic Bibliographic
- M. Ramos, Teoria da Medida, Texto de Apoio às Aulas, 2005; - Outros textos de apoio teórico/práticos a facultar pelo docente durante o trimestre;
Complementar Bibliographic
- M. Capinski, E. Kopp, Measure, Integral and Probability, Springer-Verlag, 2004 (segunda edição). - Seán Dineen, Probability Theory in Finance, Graduate Studies in Mathematics, Volume 70, AMS, 2005. - D. Williams, Probability with Martingales, Cambridge Mathematical Textbooks, 1995 (quarta edição).
