MEASURE AND INTEGRATION

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 form the basis of the course. Learning assessment consists of a written test and an oral test, both of which may be conducted electronically if necessary.

Information for Students with Disabilities and/or DSA

To ensure equal opportunities and comply with current regulations, students with disabilities and/or DSA are encouraged to request a personal interview to discuss and plan any necessary compensatory and/or dispensatory measures, tailored to their specific needs.

For assistance, students may also contact the CInAP (Centre for Active and Participated Integration – Services for Disabilities and/or DSA) representative at our department, Prof. Daniele, or the Degree Course Coordinator.

The contents of the courses of Mathematical Analysis I and II and Topology. 

Lebesgue measure. Measurable functions. Lusin's Theorem. Lebesgue integral. Fubini & Tonelli 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.

Two written exams will be administered: a midterm and a final. Students who pass both written exams are eligible to take an oral exam to complete the assessment. The evaluation will consider the following criteria: clarity of speech, depth of knowledge, and ability to connect different topics.

Students must demonstrate that they have acquired sufficient understanding of the main topics covered throughout the course and that they can successfully solve at least the simplest assigned exercises. An average of the written and oral exam grades will not be calculated.

The grading will generally follow the criteria outlined below:

Not approved: The student has not acquired the basic concepts and is unable to complete the exercises.

18–23: The student demonstrates minimal mastery of the basic concepts. Their ability to explain and connect the content is limited, and they can solve only simple exercises.

24–27: The student shows a good command of the course material. Their skills in explaining and linking content are solid, and they complete exercises with only minor errors.

28–30 cum laude: The student has fully mastered all course content and can clearly explain and critically connect the material. They solve exercises completely and without mistakes.