ISTITUZIONI DI ANALISI PER LE APPLICAZIONI

Module MODULO 1

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 deal with 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. 

Lectures and classroom exercises. The learning assessment includes a written test and an oral test. Both may also be carried out electronically, if conditions require it. 

PLEASE NOTE: Information for students with disabilities and/or DSA 

To ensure equal opportunities and in compliance with current laws, interested students may request a personal interview in order to plan any compensatory and/or dispensatory measures, based on the educational objectives and specific needs. 

You can also contact the CInAP (Centre for Active and Participated Integration - Services for Disabilities and/or DSA) contact teacher of our department, Prof. Daniele, or the President of the Degree Course.

Lebesgue measure. Measurable functions. Lusin's theorem. Lebesgue integral. Fubini's and Tonelli's theorems. 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. 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. Main properties. The problem of the passage to the limit under the integral sign. Summable p-th power functions. Holder inequality. Convergence in mean of order p. Product measures and extensions of the Fubini and Tonelli theorems. 

1. C. Miranda, Istituzioni di Analisi Funzionale Lineare, Unione Matematica Italiana, Bologna, 1978.
2. C. Pucci, Istituzioni di Analisi Superiore, Unione Matematica Italiana, Bologna, 2013.
3. R.L. Wheeden - A. Zygmund, Measure and Integral. An Introduction to Real Analysis (Second Edition), CRC Press, Boca Raton, 2015.
4. M. Muratori - F. Punzo - N. Soave, Esercizi svolti di analisi reale e funzionale, Società Editrice Esculapio, Bologna, 2021.

During the course, two written tests will be carried out in itinere, one at the halfway point and one at the end. Students who pass both are exempted from taking the full written test scheduled for each exam session. After passing the written test, an oral interview must be taken. Both for the in itinere tests and for the final exam, the following will be taken into account: clarity of exposition, completeness of knowledge, ability to connect different topics. The student must demonstrate that he/she has acquired sufficient knowledge of the main topics covered during the course, and that he/she is able to complete at least the simplest of the assigned exercises. The average between the written and oral grades is not foreseen. The following criteria will normally be followed for assigning the grade:

not approved: the student has not acquired the basic concepts and is not able to complete the exercises.

18-23: the student demonstrates a minimal mastery of the basic concepts, his/her skills in expounding and connecting the contents are modest, he/she is able to solve simple exercises.

24-27: the student demonstrates a good command of the course content, his/her skills in expounding and connecting the content are good, he/she solves the exercises with few errors.

28-30 cum laude: the student has acquired all the course content and is able to fully explain it and connect it with a critical spirit; he/she solves the exercises completely and without errors. The learning assessment may also be carried out electronically, if conditions require it.