# ELEMENTS OF ADVANCED MATHEMATICAL ANALYSIS Module MODULE 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

The course will be performed through frontal lessons. If necessary, the telematic way will be adopted. Should teaching be carried out in mixed mode or remotely, it may be necessary to introduce changes with respect to previous statements, in line with the programme planned and outlined in the syllabus. Students with disabilities and/or DSA are invited to plan, with the teacher, possible compensatory measures based on specific needs. They can also contact the referent teacher CInAP of the DMI.

## Required Prerequisites

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

## Attendance of Lessons

The partecipation in the lecture classes is strongly recommended.

## Detailed Course Content

Sequences of sets. Measurable spaces. Outer measures, measures, signed measures and generalized measures. Theorem of Carathéodory. Theorem of Jordan-Hahn. Complete measure spaces. Completion of a measure space. Absolute continuity of a set function in the sense of Vitali and in the sense of Caccioppoli. Measurable functions. Various types of convergence for a sequence of measurable functions. Theorem of Severini-Egoroff. Theorem of Weyl-Riesz. Borel sets in a topological space. Borel measures. Integration of a measurable function over a measure space. Summable functions. Properties of the integral. Passing to the limit under the integral sign. p-summable functions. Theorem of Holder-Riesz. Convergence in mean of order p. Product of two measures. Theorem of Tonelli. Theorem of Fubini.

## Textbook Information

1. E. Hewitt and K. Stromberg, Real and Abstract Analysis, Springer, 1965.

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

## Course Planning

SubjectsText References
1Measure theory (24 hours)1, teacher's notes
2Integration theory (23 hours)1, teacher's notes

## Learning Assessment

### Learning Assessment Procedures

The examen consists in an oral examination in which the student will be required to expose some definitions and some theorems (statement and proof). The assessment of learning could also be carried out by electronic means provided special circumstances should be required that. Normally, the following criteria will be applied in determing the vote of the single parts (P.C and final) of the examination:

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