# Mathematical foundations of probability

#### KORBA Anna

Department: Statistics

### 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 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

BRIANE M et PAGES G. : Théorie de l’intégration, VUIBERT, 1998 [10 BRI 00 A].
GRAMAIN A. : Intégration, HERMANN [16 GRA 00 A].