Measure theory and the Lebesgue integral

Objective

This course introduces the mathematical bases of probability theory: the theories of measure and integration according to Lebesgue.

Assessment:

The overall grade for the course will be the average of the continuous assessment (“contrôle continu”, CC) grade (50%) and the written final exam (50%).

The continuous assessment grade is made up of three elements, each graded out of twenty points: (i) the mid-term grade, (ii) the grade for attendance at tutorial sessions (TD), whose attendance is mandatory, and (iii) the grade for participation in tutorial sessions. The CC grade is calculated as follows: 50% of the mid-term grade + 25% of the attendance grade + 25% of the maximum between the participation grade and the mid-term grade.

The attendance grade is calculated according to the scale available on the school's intranet.

Planning

THEORY OF MEASURE

1. Sigma-algebras and collections of subsets -Definition. Generated sigma-algebra, inverse image sigma-algebra, products of measurable spaces.
2. Measurement, measured space -Definitions, elementary properties, characterisation of a finite measure.
3. 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.
4. Measurable applications -Definition, Borelian functions, examples, properties, measure transport, image measure, simple functions over a measurable space: definition and approximation theorem.
5. Theory of measure and probabilities

INTEGRATION

1. 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.
2. 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
3. Expected values and moments in probability

FURTHER STUDY

1. $L^p$ spaces -Definitions, properties, Holder and Minkowski inequalities, duality.
2. 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 

LE GALL J.-F. : Partie I, Intégration, Probabilités et Processus Aléatoires

REVUZ D. : Mesure et intégration, 1998: HERMANN