Mathematical Analysis 4
A.Y. 2018/2019
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 Rn.
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 Rn.
Surface integrals, relationships between integration and differentiation. Integration on oriented manifolds. Divergence theorem. Green's formula in the plane. Stokes theorem in R3.
MAT/05 - MATHEMATICAL ANALYSIS - University credits: 6
Practicals: 22 hours
Lessons: 36 hours
Lessons: 36 hours
Professors:
Messina Francesca, Peloso Marco Maria
Professor(s)
Reception:
By appointment
My office, room 1021 Dipartimento di Matematica