D1. To know the fundamental results on measure theory, integration and Lebesgue spaces.
D2. To apply the theoretical acquired skills to solve problems and exercises.
D3. To recognize the basic techniques of the covered topics for their applications to new problems.
D4. To be endowed with the competence to express the fundamental concepts with command of the language and a proper presenation.
D5. To be able to autonomously consult the specialized texts.

DIFFERENTIAL CALCULUS AND INTEGRAL CALCULUS ON R^N. METRIC SPACES. Propaedeutic exams: Analisi Matematica II.

Measure theory. Integration. Spaces of integrable functions.

E.H.Lieb, M. Loss, Analysis, American Mathematical Society, 1997
Andrew M. Bruckner, Judith B. Bruckner, Brian S. Thomson - Real analysis (1997, Prentice-Hall)
H.L. Royden, Real Analysis, MacMillan, 1968.
An Introductory Course in Lebesgue Spaces, Rene ́ Erl ́ın Castillo • Humberto Rafeiro, Springer, CMS Books in Mathematics.
HALMOS Measure Theory, Springer.
Rudin W. - Real and complex analysis-MGH (1986).
Saks S. - Theory of the integral (1937).
Terence Tao - An Introduction To Measure Theory (January 2011 Draft).

Books can be purchased online or in specialized bookstores.

Measure theory: algebras and sigma-algebras of sets. Measurement spaces. Finite and sigma-finite measurements. Complete measurements, completion of a measurement. Notion of external measurement. sigma-algebra of measurable sets and measurement generated by external measurement. Lebesgue external measure on R ^ n and Lebesgue measure. Characterization of measurable sets. Caratheodory theorem. Measurement on a semialgebra. Integration. Measurable functions. Simple functions. Almost uniform convergence. Egorov-Severini theorem. Convergence in measure, and Cauchy convergence to a degree. Approximation in measure of measurable functions on R ^ n with step and continuous functions. Lusin theorem. Integral for simple functions and measurable non-negative functions. Fatou's lemma. Monotone convergence theorem, its consequences. Integral of functions of variable sign. Dominated convergence theorem and its consequences. Absolute continuity of the integral. Derivation theorem under the integral sign. Comparison between Lebesgue integral on the line, Riemann integral and improper integrals. Lp Spaces Inequalities of Young, H¨older, Minkowsky. Convergence in Lp and convergence in measure. Density in Lp of the simple simple functions out of finite measure sets. Dual characterization of the Lp standard, various versions. Chebishev inequality. Completeness of the Lp spaces (Riesz-Fisher theorem). Integral Minkowski inequality. Interpolatory inequality. Young inequality for convolutions, nuclei amplifiers. Approximation with smooth functions in Lp (Rn).Riesz Fisher's theorem. Density results. Convolutions. Density and linear operators. Uniform convexity of L^p. Riesz representation theorem. Weak convergence. Compactness in L^p. Riesz's Frechet Kolmogorov theorem.

Frontal lessons and exercises. Active involvement of the students. Homework available on the web site.
Group exercises and activities.

Detailed information on the course's MS Teams site. Any changes to the arrangements described here that become necessary to ensure the implementation of safety protocols related to the COVID19 emergency will be communicated on the Department, Course of Study and teaching website."

Oral examinations. The oral examination aims to access the students’ knowledge of the theoretical and the applicative aspects of the covered topics.

Quality education.