Mathematical Analysis 4

A.Y. 2020/2021
Overall hours
Learning objectives
The aim of the course is:
- to complete the set of the basic techniques of the integral calculus in more real variables;
- to provide basic notions on measure theory, with special application to the Lebesgue measure in Rn.
Expected learning outcomes
Knowledge of the topics and results, and application to exercises that need also computational techniques.
Course syllabus and organization

Single session

Lesson period
Second semester
The lectures and problem sessions (esercitazioni) will be conducted remotely
in virtual classrooms (on the zoom platform) via live streaming, allowing for real-time interaction between the students and the instructor.
Course syllabus
-- Lebesgue measure and integration: Lebesgue measure, measurable functions and the Lebesgue integral in Euclidean spaces. Passage of limits under the integral sign and integration of series of functions. Comparison between the integrals of Lebesgue and Riemann. The reduction theoremsof Fubini and Tonelli. Change of variables formulas. Integrals depending on parameters.

-- Abstract measures and integration: abstract measures, outer measures and metric outer measures, measurable functions, integration with respect to a (positive) measure.

-- Hausdorff measure and integration: Hausdorff outer measure and measure, comparison between Lebesgue outer measure and Hausdorff n-dimensional outer measure in n-dimensional Euclidian space. Hausdorff measure and Lipschitz maps. Integration with respect to Hausdorff measure, integration on parameterized hypersursurfaces, graphs and manifolds.

-- The fundamental theorems of integral calculus in several variables: regular open sets and the fundamental theorem of calculus, integration by parts nd the divergence theorem. Generalizations and applications.
Prerequisites for admission
Mathematical Analysis 1, 2 and 3.
Teaching methods
Traditional blackboard lectures. Attendance strongly suggested.
Teaching Resources
K.R. Payne - Misura ed Integrazione (available online at the course webpage)
E. Lanconelli - Lezioni di Analisi Matematica 2, Prima Parte (2° edizione) 2000, e Seconda Parte 1997, Pitagora Editrice, Bologna
E. Giusti - Analisi Matematica 2, 3° edizione, Bollati Boringhieri, Torino 2003
G. Molteni e M. Vignati - Analisi Matematica 3, Città Studi Edizioni, Milano 2006
E. Giusti - Esercizi e Complimenti di Analisi Matematica, 2° Volume, Bollati Boringhieri, Torino 1992
C. Maderna, G. Molteni e M. Vignati - Esercizi Scelti di Analisi Matematica 2 e 3, Città Studi Edizioni, Milano 2006
Assessment methods and Criteria
The final examination consists of two parts: a written exam and an oral exam.

- During the written exam, the student must solve some exercises in the format of open-ended and/or short answer questions, with the aim of assessing the student's ability to solve problems in measure theory, interchanging the operations of integration and limits, and integration in several variables. The duration of the written exam will be proportional to the number of exercises assigned, also taking into account the nature and complexity of the exercises themselves (however, the duration will not exceed three hours). In place of a single written exam given during the first examination session, the student may choose instead to take 2 midterm exams. The outcomes of these tests will be available in the SIFA service through the UNIMIA portal.
- The oral exam can be taken only if the written component has been successfully passed. In the oral exam, the student will be required to illustrate results presented during the course and will be required to solve problems in order to evaluate her/his knowledge and comprehension of the arguments covered as well as the capacity to apply them.

The complete final examination is passed if both the written and oral parts are successfully passed. Final marks are given using the numerical range 0-30, and will be communicated immediately after the oral examination.
MAT/05 - MATHEMATICAL ANALYSIS - University credits: 6
Practicals: 24 hours
Lessons: 36 hours
Professors: Payne Kevin Ray, Rondi Luca
Educational website(s)