Random phenomena are modelled using modern probability theory, defined in the 1930s by Kolmogorov using measure theory as a cornerstone.  This course aims to provide a deep understanding of this theory. It is indeed an asset to forge intuition, to understand the objects involved and to mobilize them in an applied or theoretical framework.

This course is designed for an audience with a variety of interests: it may be of interest to students wishing to deepen their study of probability theory on the one hand, and on the other hand it may be of interest to students who intend to use it in business applications (a good understanding of probability theory is essential in order to be able to orient oneself in the world of applications and to innovate there).

Each week is devoted to a theme of measure theory, with applications related to probability, involving discussions around exercises. The last session is devoted to oral presentations.

• Week 0: Introduction, description of the course
• Week 1: Measures. (σ-field, generated σ-field, mesures, monotone class theorem)
• Week 2: Measurable functions. (measurable functions, product σ-field, approximation by simple functions, Doob-Dynkin lemma, constructing measures and Stieltjes measures)
  
• Week 3: Integration. (integral with respect to a measure, monotone convergence, Radon-Nikodym theorem, Fubini theorem, convergence theorems)
  
• Week 4: Independence (Independence of σ-fields and of random variables, transfer theorem, independence and integration, dummy function method, independence of infinite families, Kolmogorov 0-1 law, Borel-Cantelli lemmas) and recap on different notions of convergence for random variables.
  
• Week 5: Conditioning. (conditional expectation, conditional law, Gaussian vectors)
  
• Week 6: Brownian motion (1/2). (construction)
  
• Week 7: Brownian motion (2/2). (Simple and strong Markov properties)
  
• Week 8: Oral presentations.

The evaluation is based on an oral presentation by pairs of a research or overview article on a model, which is attended by all students. The objective is both individual (to learn to read a primary source and to present its content orally in English in a given time) and collective (to see a variety of models and applications in probability). 

The course is delivered in english.