Mathematical Foundations of Probability Theory
Teacher
BUTUCEA Cristina
Department: Statistics
ECTS:
4
Course Hours:
21
Tutorials Hours:
21
Language:
French
Objective
This course introduces the mathematical bases of probability theory: the theories of measure and integration according to Lebesgue.
Planning
THEORY OF MEASURE
- Sigma-algebras and collections of subsets -Definition. Generated sigma-algebra, inverse image sigma-algebra, products of measurable spaces.
- Measurement, measured space -Definitions, elementary properties, characterisation of a finite measure.
- Extension of a measure with applications -Extended theorem, outer measure, Borel measure, negligible sets, sigma algebra and completed measure, sigma algebra and Lebesgue measure, finite product of a family of measured spaces.
- Measurable applications -Definition, Borelian functions, examples, properties, measure transport, image measure, simple functions over a measurable space: definition and approximation theorem.
- Theory of measure and probabilities
INTEGRATION
- Integration of positive measurable functions -Integral of a simple function and of a measurable function, properties, monotone convergence theorem (Beppo Levi), Fatou's lemma, density measures, variable change theorem, the Fubini-Tonelli theorem.
- Integration of any functions -Integral of any function, $L^p$ spaces, properties, dominated convergence theorem, applications (continuity and derivative below the sum sign), Fubini's theorem, convolution
- Expected values and moments in probability
FURTHER STUDY
- $L^p$ spaces -Definitions, properties, Holder and Minkowski inequalities, duality.
- Fourier transform
References
ANSEL J.-P., DUCEL Y. : Exercices corrigés en théorie de la mesure et de l'intégration, 2015, Paris:Ellipses
BRIANE M et PAGES G. : Analyse, Théorie de l’intégration, 2012: VUIBERT
GALLOUET T. et HERBIN R. : Mesure, intégration, probabilités, 2013, Ellipses
GRAMAIN A. : Intégration, HERMANN
REVUZ D. : Mesure et intégration, 1998: HERMANN