Measure Theory and probability

Presentation

Program (detailed contents):

-Sigma-algebra

-Measure

-Measurable function, random variable

-Integration, Expectation

-Almost everywhere (almost sure) convergence

-Dominated convergence, Fatou, Beppo-Levi

-L1 convergence

-Continuity and differentiability inversion with integrals

-Fubini theorem

-Transport theorem, law of a random variable

-Lp spaces : norms, Hölder, completness

-Markov and Chebyshev inequalities

- Convolution product

Main difficulties for students:

This is a theoretical course which does not require advanced knowledges. This course is at the heart of many developments in modern mathematics.

Objectives

At the end of this module, the student will have understood and be able to explain (main concepts):

The definition of measures and integration. The limit theorems (Fatou, Lebesgue, Beppo-Levi). Lp spaces and related norms, convolution product. The rigorous definition of the basic notions of probability in a measure theoretic framework.

The student will be able to:

Switch a limit (or a derivative) and an integral sign, use the notion of measure, prove the different kind of convergence (almost everywhere, Lp, etc), discuss the belonging to a function in Lp, use Cauchy-Schwarz and Hölder inequality, compute a convolution product.

Needed prerequisite

Basic knowledge on analysis (basic derivation and integration), Basic knowledge in Probability (law,expectation,variance). Basic knowledge in mathematic and theory (manipulation of “there exists”, “forall”,...)

Form of assessment

The evaluation of outcome prior learning is made as a continuous training during the semester. According ot the teaching, the assessment will be different: as a written exam, an oral exam, a record, a written report, peers review...