Measure and Integration
Academic Year 2026/2027 - Teacher: SALVATORE ANGELO MARANOExpected Learning Outcomes
The primary objective of the course is to provide students with an in-depth understanding of the concepts and key results inherent in the theory of measure and integration in abstract spaces. This will enhance their knowledge of mathematical analysis and provide a useful tool for studying issues arising from other courses (probability, mathematical physics, etc.). This will be achieved by initially examining the theory of measure and integration in the simple context of Euclidean spaces. In this case, it is a rich theory, closely related to known facts of infinitesimal calculus and generalizing them. Specifically, the course aims to achieve the following objectives:
Knowledge and understanding
The main topics inherent in the theory of measure and integration according to Lebesgue will be reviewed, also with the aim of deepening and unifying some notions and methodologies learned in previous mathematical analysis courses. The course will then examine the theory of measure and integration in abstract spaces, with particular attention to the various types of convergence for sequences of measurable functions, the problem of limit transition under the integral sign, series integration, and iterated integration.
Applying knowledge and understanding
Students will learn to solve Lebesgue integrals (simple or multiple), be able to study the various types of convergence for a sequence (or series) of functions, and determine when limit transition under the integral sign or series integration is permitted. They will also be able to tackle simple exercises in measure theory in abstract spaces.
Making judgments
At the end of the course, students will be able to identify the most suitable abstract framework for calculating a given integral, identify the type of convergence for a given sequence (or series) of functions, and study the problem of limit transition under the integral sign or series integration. They will also be able to judge which of the basic concepts of mathematical analysis naturally extend to real-world analysis.
Communication skills
During classes, students will be constantly encouraged to contribute, expressing their perspectives on both theoretical topics and applications. This is intended to develop their critical thinking and intuition, as well as accustom them to communicating in mathematically correct language.
Learning skills
They will be stimulated and periodically tested with in-class exercises and simple theoretical questions to be developed individually.
Course Structure
Lectures and exercises in the classroom.
Verification of learning involves a written test and an oral test. Both can also be carried out electronically, if conditions will require it.
PLEASE NOTE: Information for students with disabilities and / or DSA
To guarantee equal opportunities and in compliance with the laws in force, interested students can ask for a personal interview in order to plan any compensatory and / or dispensatory measures, based on the didactic objectives and specific needs.
It is also possible to contact the referent teacher CInAP (Center for Active and Participated Integration - Services for Disabilities and / or SLD) of our department, prof. Daniele, or the President of the master degree..
Required Prerequisites
The contents of the courses Mathematical Analysis I and II and Topology.
Attendance of Lessons
Detailed Course Content
Review of Lebesgue measure. Sequences of sets. Measurable spaces. Measures, relative measures, and generalized measures. Jordan-Hahn theorem. Complete mensural spaces. Completion of a mensural space. Absolute continuity according to Vitali or Caccioppoli of a set function. Measurable functions, Lusin's theorem, approximation of a measurable function by simple functions. Various types of convergence for sequences or series of measurable functions. Severini-Egoroff and Weyl-Riesz theorems. Borel measures. Integration of a measurable function on a mensural space. Summable functions and their main properties. The problem of the transition to the limit under the integral sign. Beppo Levi and Lebesgue theorems, Fatou's lemma. Equally and absolutely continuous set functions, Vitali's theorem. Summable p-th power functions. Holder and Minkowski inequalities. Convergence in mean of order p. Measure and integration in product spaces, Fubini and Tonelli theorems.
Textbook Information
- C. Pucci, Istituzioni di Analisi Superiore, Unione Matematica Italiana, Bologna, 2013.
- A. Tesei, Istituzioni di Analisi Superiore, Bollati Boringhieri, Torino, 1997.
- R.L. Wheeden - A. Zygmund, Measure and Integral. An Introduction to Real Analysis (Second Edition), CRC Press, Boca Raton, 2015.
- M. Muratori - F. Punzo - N. Soave, Esercizi svolti di analisi reale e funzionale, Società Editrice Esculapio, Bologna, 2021.
Course Planning
| Subjects | Text References | |
|---|---|---|
| 1 | Lebesgue measure and Integral. | 3) |
| 2 | Abstract measure and integral. | 3) |
| 3 | Sequences of measurable functions. | 3) |
| 4 | Passage to the limit under the integral sign. | 3) |
| 5 | p-th power summable functions. | 3) |
Learning Assessment
Learning Assessment Procedures
Two ongoing tests will be administered during the course, one midway through the course and one at the end. Students who pass both tests are exempt from taking the full exam scheduled for each exam session. For both the midterm tests and the final exam, the following will be taken into consideration: clarity of presentation, completeness of knowledge, and the ability to connect different topics. Students must demonstrate sufficient knowledge of the main topics covered during the course and the ability to complete at least the simplest of the proposed exercises.
Grades will generally be assigned according to the following criteria:
Failed: The student has not mastered the basic concepts and is unable to complete the exercises.
18-23: The student demonstrates minimal mastery of the basic concepts, their presentation and connection skills are modest, and they can solve simple exercises.
24-27: The student demonstrates a good command of the course content, good presentation and connection skills, and solves the exercises with few errors.
28-30 cum laude: The student has mastered all the course content and is able to present it thoroughly and connect it critically; he or she solves the exercises completely and without errors.
Assessment may also be conducted online, if conditions require.
Examples of frequently asked questions and / or exercises
- Generalized measures and their main properties.
- Relative measure and the Jordan-hahn theorem.
- Measurable functions and integration in abstract spaces.
- Various types of convergence for sequences of measurable functions.
- The problem of passing to the limit under the integral sign.
- Convergence in mean of order p. Holder and Minkowski inequalities.
- Product measures, Fubini's and Tonelli's theorems.