MEASURE AND INTEGRATION
Academic Year 2025/2026 - Teacher: SALVATORE ANGELO MARANOExpected Learning Outcomes
The main objective of the course is to provide the student with an in-depth treatment of the concepts and main results inherent in the theory of measure and integration in abstract spaces, in order to enrich their cultural background in the field of mathematical analysis and to provide a useful tool for the study of issues coming from other teachings (harmonic analysis, probability calculus, 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, which has a close relationship with known facts of infinitesimal calculus and which generalizes them. In particular, the course has the following objectives:
Knowledge and understanding: the main topics inherent in the theory of measure and integration according to Lebesgue will be studied first, also with the aim of deepening and unifying some notions and methodologies learned in previous courses of mathematical analysis. The theory of measure and integration in abstract spaces will then be examined, with particular attention to the various types of convergence for sequences of measurable functions, the problem of the passage to the limit under the integral sign, and iterated integration.
Applying knowledge and understanding: the student will learn to solve Lebesgue integrals (simple or multiple), will be able to study the various types of convergence for a sequence (or series) of functions and say when the passage to the limit under the integral sign is permitted, as well as tackle easy exercises in measure theory in abstract spaces.
Making judgements: at the end of the course the student will be able to identify the most suitable abstract scope for calculating a given integral, identify the type of convergence of a given sequence (or series) of functions, study the problem of the passage to the limit under the integral sign. He will also be able to judge which of the basic concepts of mathematical analysis extend naturally to real analysis.
Communication skills: during the lessons, students will be constantly invited to intervene, expressing their point of view, both on theoretical topics and on applications. This is aimed at developing their critical sense and intuition, as well as getting them used to communicating with mathematically correct language.
Learning skills: they will be stimulated and periodically verified with classroom 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..
Attendance of Lessons
Detailed Course Content
Lebesgue measure. Vitali's covering lemma. Measurable functions. Lusin's theorem. Lebesgue integral. Fubini's and Tonelli's theorems. Sequences of sets. Measure spaces. Measures, relative measures and generalized measures. Jordan-Hahn's theorem. Complete mensural spaces. Completion of a mensural space. Absolute continuity according to Vitali or Caccioppoli of a set function. Measurable functions. Various types of convergence for sequences (or series) of measurable functions. Severini-Egoroff's and Weyl-Riesz's theorems. Borel measures. Integration of a measurable function on a mensural space. Summable functions. Main properties. The problem of the passage to the limit under the integral sign. p-th power summable functions. Holder's inequality. Convergence in mean of order p. Product measures and extensions of Fubini's and Tonelli's theorems
Textbook Information
- C. Miranda, Istituzioni di Analisi Funzionale Lineare, Unione Matematica Italiana, Bologna, 1978.
- C. Pucci, Istituzioni di Analisi Superiore, Unione Matematica Italiana, Bologna, 2013.
- 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
During the course, there will be two written tests in progress, one in the middle of the course and one at the end. Students who pass both are exempted from taking the complete written test required for each session. After the written test, an oral interview must be taken.
Both for the ongoing tests and for the final exam, the following will be taken into account: clarity of presentation, completeness of knowledge, ability to connect different topics. The student must demonstrate that he has acquired sufficient knowledge of the main topics covered during the course, and that he is able to carry out at least the simplest of the assigned exercises. There is no average between the written and oral grades. The following criteria will normally be followed to assign the grade:
not approved: the student has not acquired the basic concepts and is not able to carry out the exercises.
18-23: the student demonstrates minimal mastery of the basic concepts, his skills in exposition and connection of contents are modest, he is able to solve simple exercises.
24-27: the student demonstrates good mastery of the course contents, his presentation and content connection skills are good, he solves the exercises with few errors.
28-30 cum laude: the student has acquired all the contents of the course and is able to explain them fully and connect them with a critical spirit; she solves the exercises completely and without errors.
Verification of learning can also be carried out electronically, should the conditions require it.