Elements of Infinitesimal Analysis IV

Cod: 21033

Department: DCET

Department: DCET

ECTS: 6

Scientific area: Mathematics

Scientific area: Mathematics

Total working hours: 156

Total contact time: 26

Total contact time: 26

Every day one deals with measure quantities, e.g., lenghts, velocities, temperatures. Underlying this fact one has the notion of a measure. The definition of a measure and its main properties are studied in this course unit. In particular, the integral calculus already known from previous course units on Calculus is generalized to general measures.

Measures

Integrals

Integrals

At the end of this course students are expected to be able to apply the notions and some of the main results of measure theory.

1. The Riemann integral and the Riemann-Stieltjes and the Darboux-Stieltjes integrals

2. Notion of measure: definition and examples. Lebesgue measure

3. Definition and construction of the integral

4. Properties and convergence results

5. Relation between the Lebesgue integral and the Riemann integral

6. Integration over product spaces: definition and Fubini's theorems

7. Absolutely continuous measures

2. Notion of measure: definition and examples. Lebesgue measure

3. Definition and construction of the integral

4. Properties and convergence results

5. Relation between the Lebesgue integral and the Riemann integral

6. Integration over product spaces: definition and Fubini's theorems

7. Absolutely continuous measures

Course materials will be provided online

**Complementary Bibliography: **

Magalhães, L. T.,*Integrais Múltiplos*, Texto Editora, 1993.

Capinski, M., Kopp, E.,*Measure, Integral and Probability*, 2^{nd} edition, Springer, 2005.

Magalhães, L. T.,

Capinski, M., Kopp, E.,

E-Learning.

Continuous assessment is privileged: 2 or 3 digital written documents (e-folios) during the semester (40%) and a
presence-based final exam (p-folio) in the end of the semester (60%). In due time, students can alternatively choose to perform one
final presence-based exam (100%).

Students are recommended to have previously knowledge in the Elements of Infinitesimal Analysis II and III curricular units.