MEASURE AND INTEGRATION

The aim of the course is to make the students familiar with basic concepts, main theorems and most used techniques in Measure and Integration Theory. This will give the students a more complete education in the field of Mathematical Analysis and will provide them with useful prerequisites in order to follow more advanced courses.

The course main topics will be explained by the teacher during formal lectures. These will be focusing on each topic's general principles and new concepts that have not been studied before. Each's topic's additional resources and subchapters will be presented by turning over groups of students. The goal is to have students develop study autonomy and teaching abilities, skills that are essential for students who want to pursue a career in research or teaching.

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.

Lebesgue measure. Measures, outer measure and Carathéodory's theorem. Borel sets of a topological space. Borel measures and distribution functions.Completion of a measure space. Measurable funcions. Sets which are not Lebesgue measurable and Lebesgue measurable sets that are not Borel sets. Signed measures. Integration in a measure space. L^p-spaces. Various types of convergence of sequences of measurable functions. Product measure and Fubini's theorem.

1. A. Villani, Appunti del corso di Istituzioni di Analisi Superiore, lecture notes on line

2. W. Rudin, Real and Complex Analysis, Third edition, Mc Graw Hill