# ISTITUZIONI DI ANALISI PER LE APPLICAZIONIModule MODULO 1

**Academic Year 2023/2024**- Teacher:

**Biagio RICCERI**

## Expected Learning Outcomes

The main goal of the course is to provide the student with a more in depth treatment of the most important concepts and results within the abstract theory of measure and integration. In such a way, the student will enrich his/her cultural background in the field of Mathematical Analysis and will acquire useful tools to follow other courses.

In more detail, following the Dublin descriptors, the objectives are the following:

Knowledge and understanding: the student will learn to work with the most typical concepts and techniques of the abstract theory of measure and integration.

Applying knowledge and understanding: the student will be guided in the ability to realize applications of the general results gradually established.

Making judgements: the student will be stimulated to study autonomously some results not developed during lessons.

Communication skills: the student will learn to expose in a clear, rigorous and concise manner.

Learning skills: the student will be able to face exercices and found proofs of simple results.

## Course Structure

## Required Prerequisites

## Attendance of Lessons

The partecipation in the lecture classes is strongly recommended.

## Detailed Course Content

## Textbook Information

Some teacher's notes will be published on the Studium page of the course.

## Course Planning

Subjects | Text References | |
---|---|---|

1 | Measure theory (24 hours) | 1, teacher's notes |

2 | Integration theory (23 hours) | 1, teacher's notes |

## Learning Assessment

### Learning Assessment Procedures

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

18-23: the student shows a minimal mastery of the basic concepts, his/her exposure and linking skills are modest, he/she is able to solve simple exercises.

24-27: the student shows a good mastery of the basic concepts, his/her exposure and linking skills are good, he/she solves exercises with a few mistakes.

28-30 cum laude: the student has acquired all the course contents and is able to expose and connect them in a complete and critic way, he/she solves exercices completely and without mistakes.

### Examples of frequently asked questions and / or exercises

Theorem of Jordan-Hahn

Completion of a measure space

Characterization of the absolute continuity of a set function

Theorem of Severini-Egoroff

Theorem of Weyl-Riesz

Theorems on passing to the limit under the integral sign

Characterization of the convergence in mean of order p

Theorem of Tonelli

Theorem of Fubini