Learning outcomes

The first part of the course presents the main results of the Measure and Integration theory within the framework of measured spaces. The second part deals with the theory of the Fourier transform of Lebesgue integrable functions, as well as the Laplace transform.

Goals

The objective is to introduce to the theory of the Lebesgue integral, and to show the main contributions of this theory compared to the Riemann integral.

Content

The table of contents consists of ten chapters: Introduction. Measurable space and measure function. Measurable functions. Integral of a non-negative measurable function. Integrated functions. Lp spaces. Lebesgue measurement on R. Product measurement. Fourier transform in L1. Fourier transform in L2. La place transform.
 

Table of contents

The table of contents consists of ten chapters: I. Introduction. II. Measure and measured space. III. Measurable functions. IV. Integral of a non-negative measurable function. V. Integrable functions. VI. Lp spaces. VII. Lebesgue measurement on R. VIII. Product space. IX. Fourier transform in L1. X. Fourier transform in L2. Laplace transform.
 

Exercices

Illustration of the concepts and results of the course.

Assessment method

The evaluation has two parts. 1) An individual oral exam which aims to assess the student's knowledge and level of understanding of the course (definitions, statements and demonstrations of theorems, summary questions ...). 2) A written exercise exam which aims to test the student's ability to apply the results seen in the course. If the student obtains a mark higher than 10/20 in the written and oral exams, the final mark is the arithmetic mean of the marks. Otherwise, the final dimension is the integer part of the geometric mean of the dimensions, with at least 7/20 or more for each part of the exam (=> minimum of both marks).
 

Sources, references and any support material

"Measure Theory - A first Course", Carlos S. Kubrusly, Elsevier, 2007

"Measure Theory", Paul R. Halmos, Springer, 1974

"Measure Theory", J.L. Doob, Springer, 1993

 

Language of instruction

Anglais
Training Study programme Block Credits Mandatory
Bachelor in Mathematics Standard 0 6
Bachelor in Mathematics Standard 3 6