Mathematics

EMANUELE NUNZIO SPADARO
EMANUELE NUNZIO SPADARO

General targets: to acquire basic knowledge in measure and integration theory, spaces L^p, Fourier series.

Specific targets: to acquire the ability to use the definitions and theorems contained in the course.

Knowledge and understanding: the student will have acquired the basic notions and results related to the Theory of Abstract Measure, to the construction of the Lebesgue Measure, to the Theory of Integration, to the spaces L^p, to the spaces of Hilbert, to the Fourier series.

Apply knowledge and understanding: the student will be able to understand the concept of measurement and integral in abstract spaces, to integrate discontinuous functions, to operate with different notions of convergence in L^p, to use the Fourier series in L^2.

Critical and judgmental skills: the student will have the basis to deal with some problems of pure and applied mathematics, related to the Theory of Measurement, the spaces L^p, the spaces of Hilbert, the Fourier series.

Communication skills: the student will be able to expose the course contents in a clear and understandable way, both in the written and oral part.

Learning skills: the knowledge acquired will allow study, individual or taught in a master-level course, relating to more specialized aspects concerning the Theory of Measurement, the spaces L ^ p, the spaces of Hilbert, the Fourier series.

The student will have acquired the basic notions and results related to the Theory of Abstract Measure, to the construction of the Lebesgue Measure, to the Theory of Integration, to the spaces L^p, to the spaces of Hilbert, to the Fourier series.
The student will be able to understand the concept of measurement and integral in abstract spaces, to integrate discontinuous functions, to operate with different notions of convergence in L^p, to use the Fourier series in L^2.

We require basic knowledge of calculus in one and more variables. In particular: limits, sequences and series, gradient of functions, Riemann integral, sequences and series of functions, point-wise convergence, local and uniform convergence. Linear algebra and topology.
Not required the students to have passed specific exams in advance.

Measure theory: Lebesgue measure in Rd; General measure and integration theory; Riemann and Lebesgue integral; change of variable formula; product measures, Fubini theorem; L^p spaces. Hilbert spaces and Fourier series.

H.L. Royden: “Real Analysis”.
W. Rudin: “Real & Complex Analysis”.
E. Stein, R. Shakarchi: “Real Analysis”.
E. Spadaro "Lezini di Analisi Reale".

Exercise sheets will be assigned regularly during the course.

Frontal lectures

In room, 7 hours per week
Following the course is not mandatory, but strongly encouraged.

The exams need to evaluate the learning through a written text (consiting in solving problems of the same type as those which have been proposed during the lectures) and an oral interview (consisting in a discussion about the most relevant topics of the lectures).
The duration of the written text is about three hours and can be overcome by passing two intermediate texts, both of two hours, the first one at the middle of the course and the second one immediately at the end. The first text will be centered on Measure and Integration Theory, while the second one will be concerned on Fourier series and Complex Variable.
In order to pass the exam one needs to get a score greater or equal to 18/30. The student needs to demonstrate that he/she got a sufficient knowledge of the topics of all the three parts of the program and to be able to solve at least the simplest exercises.
In order to get a score of 30/30 cum laude, the student needs to demonstrate that he/she got an excellent knowledge of all the topics of the lectures and to be able to connect them in a coherent and logical way.

- State Fatou's Theorem; explain some applications of it.
- Establish if a given set in R^d is Lebesgue measurable and compute its measure.
- State and prove Egorov Theorem.
- Put in relationship convergence a.e, a.u., in measure and in L^p for sequences of measurable functions.