Master French language at the level corresponding to the last year of the secondary school education at least.
At the end of this learning unit, the student is able to : | |
| 1 | Contribution of the course to learning outcomes in the Bachelor in Mathematics programme. By the end of this activity, students will have made progress in: - knowing and understanding a fundamental subject of the mathematics.
He/her will have developed in particular his/her capacity in: -- choosing and using methods and fundamental tools of calculation to solve mathematical problems ; -- recognizing the fundamental concepts of certain current mathematical theories ; -- establishing the main links between those theories, explaining them and motivating them by examples ; --identifying the unifying aspects of different situations and phenomena in the mathematical science, thanks to the abstract and experimental approach peculiar to the exact sciences. ¿ showing abstraction and critical mind.
He/her will have developed in particular his/her capacity in:
-- reasoning within the framework of the axiomatic method. -- Recognizing the key arguments and the structure of a mathematical proof. -- Structuring and drafting a mathematical proof in a autonomous way. -- Assessing the rigor of a mathematical reasoning and revealing its possible weaknesses. -- making the distinction between the intuition of the validity of a result and the various levels of rigorous understanding of the same result.
Learning outcomes specific to the course. By the end of this activity, students will be able to:
- retrace the historic evolution of the mathematical concepts introduced during the course and the their associated problems. - master, in an abstract context, the fundamental theorems of integration learned during the previous courses of analysis for the case of the Euclidean spaces, harmonizing the latter with the example of the outer Lebesgue measure. - Master the techniques of proof peculiar to measure theory. - Recognize the presence of certain algebraic structures in class of subsets. - Build a measure starting from a countable additive set function defined on a semi-algebra of subsets or starting from a sequence of suitably chosen measures. - Integrate a measurable function with respect to a measure. - Use the Convergence Theorems. - Know the main connections between the notions of measure and probability. |
The contribution of this Teaching Unit to the development and command of the skills and learning outcomes of the programme(s) can be accessed at the end of this sheet, in the section entitled “Programmes/courses offering this Teaching Unit”.
Within the framework of the course, the following contents are approached.
- A starting extension of the length of an interval.
- The Lebesgue's measure on R.
- The Carathéodory criterion of measurability.
- A measure on a sigma algebra of subsets.
- The Stieltjes-Lebesgue-Radon measures.
- Measurable functions.
- The integral of a measurable function with respect to a measure.
- Convergence theorems.
- L^p spaces.
- Product measures.
- Fubini's and Tonelli's Theorems.
- Convergence of a sequence of measures.
- Connections with Probability Theory.
The two activities are given in presential sessions.
Syllabus disponible sur iCampus.
