Measure and Integration Course and Application Exercises

Abstract

This course, designed for third-year undergraduate students in Mathematics (LMD sys- tem), covers the core of the material on Measure and Integration. It includes lectures with examples, and application exercises with solutions are provided at the end of each chapter to allow students to test their knowledge and prepare for tests and nal exams. Based on my experience teaching this subject for several years, I decided to prepare this course pack, which contains all the fundamental concepts related to this topic. It should be noted that the content of this course pack is exactly the same as that o¤ered in the o¢ cial curriculum, following the framework provided by the Ministry and currently applied in all departments of Algerian universities. We assume that the reader has a good knowledge of the usual topology of R, the rst principles of set theory and the concept of integration in the Riemann sense. Since this course material is a lecture, we have chosen to prove almost all results comple- tely, that is, without referring back to a well-known result or admitting a di¢ cult auxiliary result. We have also included a considerable number of solved exercises as they were tes- ted in tutorials or used in re ective assignments or knowledge assessments. It goes without saying that the reader will bene t from trying to solve the problems without rst reading the solution. The chapters of this course material conclude with solved exercises drawn from the tutorial series prepared by the teaching sta¤of the Mathematics Department at Abdelhamid Ibn Badis University of Mostaganem. The originality of this handout lies in its content, shamelessly inspired by existing lite- rature. Let s now turn to a more precise description of what you will nd in this handout. In the rst chapter, we will brie y present the useful properties concerning operations on sets, countability, limits of sets, and characteristic functions of sets. We will then introduce the notion of a sigma-algebra, particularly the Borel sigma-algebra. We will o¤er a detailed study of positive measure, exterior measure, and in particular, Lebesgue measure on the Borel sigma-algebra. The second chapter contains the general properties of measurable functions, notably measurable numerical maps, which will be denoted by L0. We will study convergence almost everywhere and convergence in measure. In the third chapter, we will address and discuss the concept of integration with respect to a positive measure. First, we will study it for measurable real-valued functions and present the Monotone Convergence Theorem (or Beppo-Levi Theorem) and its consequences. We will then examine the integral of a measurable real-valued function and conclude with a comparison of the Lebesgue integral with the Riemann integral. Finally, we will provide a general overview of the construction of the L1 space and the Dominated Convergence Theorem in this space. In the fourth chapter, we devote ourselves to the study of product measure, including Fubini s Theorems and some applications. Finally, given the frequent and recurring errors in exam papers for this subject, I ve noticed that most students don t pay enough attention to the lecture material and instead work on exercises based solely on the solutions. I therefore advise students to rst read the lecture notes carefully, then complete all the examples provided after each solution, and nally, proceed to solve the exercises without referring back to the solutions. The solutions are only useful for assessing the student s level of e¤ort. Ultimately, I hope this document will help students who want to master this area ofma- thematics.