 # 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...