Mathematical Analysis 4
A.Y. 2019/2020
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.
- 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.
Lesson period: Second semester
Assessment methods: Esame
Assessment result: voto verbalizzato in trentesimi
Single course
This course cannot be attended as a single course. Please check our list of single courses to find the ones available for enrolment.
Course syllabus and organization
Single session
Responsible
Lesson period
Second semester
Course syllabus
Positive measures and abstract integration. Measurable spaces and functions, positive measures, completion of measures. Integration with respect to a measure. Integrating sequences: monotone convergence, Fatou's lemma, dominated convergence. Normed space L1, its completeness. Product measures, Fubini's and Tonelli's theorems. Radon-Nikodym theorem.
Lebesgue measure. Lebesgue measure (and integral) in Rn (n ≥ 1), comparison with the classical theory. Cantor sets, essential pathologies. Integrals depending on a parameter, the Euler's Γ function. Riemann-Stieltjes and Lebesgue-Stieltjes integral. Differentiation and integration. Basic ideas on the Hausorff measure in
Surface integrals, relationships between integration and differentiation. Integration on oriented manifolds. Divergence theorem. Green's formula in the plane. Stokes theorem in R3.
Lebesgue measure. Lebesgue measure (and integral) in Rn (n ≥ 1), comparison with the classical theory. Cantor sets, essential pathologies. Integrals depending on a parameter, the Euler's Γ function. Riemann-Stieltjes and Lebesgue-Stieltjes integral. Differentiation and integration. Basic ideas on the Hausorff measure in
Surface integrals, relationships between integration and differentiation. Integration on oriented manifolds. Divergence theorem. Green's formula in the plane. Stokes theorem in R3.
Prerequisites for admission
Analisi Matematica 1, 2, and 3.
Teaching methods
Traditional blackboard lectures. Attandance strongly suggested.
Teaching Resources
-M. Peloso: [Course notes]
-W. Rudin: Real and complex analysis, McGraw-Hill;
-B. Gelbaum, J. Olmsted: Counterexamples in analysis, Dover Publications Inc., Mineola, NY, 2003;
-H.L. Royden: Real Analysis, MacMillan publ. co..
-G. Molteni, M. Vignati: Analisi Matematica 3, Città Studi Edizioni, Milano, 2006;
-E. Lanconelli: Lezioni diAnalisi Matematica 2 (seconda parte), Pitagora Editrice, Bologna, 1997.
-W. Rudin: Real and complex analysis, McGraw-Hill;
-B. Gelbaum, J. Olmsted: Counterexamples in analysis, Dover Publications Inc., Mineola, NY, 2003;
-H.L. Royden: Real Analysis, MacMillan publ. co..
-G. Molteni, M. Vignati: Analisi Matematica 3, Città Studi Edizioni, Milano, 2006;
-E. Lanconelli: Lezioni diAnalisi Matematica 2 (seconda parte), Pitagora Editrice, Bologna, 1997.
Assessment methods and Criteria
The final examination consists of three parts: a written exam, an oral exam and a lab 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 operation of integration and limit, 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.
- 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 operation of integration and limit, 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: 22 hours
Lessons: 36 hours
Lessons: 36 hours
Professors:
Messina Francesca, Peloso Marco Maria
Shifts:
Professor(s)
Reception:
By appointment
My office, room 1021 Dipartimento di Matematica