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### Measure and integration

 Code: 45779 ECTS: 5.0 Lecturers in charge: doc. dr. sc. Rudi Mrazović doc. dr. sc. Vanja Wagner Lecturers: Adrian Beker - Exercises Aleksandar Bulj , mag. math. - Exercises dr. sc. Petra Lazić - Exercises English level: 1,0,0 All teaching activities will be held in Croatian. However, foreign students in mixed groups will have the opportunity to attend additional office hours with the lecturer and teaching assistants in English to help master the course materials. Additionally, the lecturer will refer foreign students to the corresponding literature in English, as well as give them the possibility of taking the associated exams in English.

### 1. komponenta

Lecture typeTotal
Lectures 30
Exercises 30
Description:
COURSE AIMS AND OBJECTIVES: To introduce notions of measure, measurable functions and abstract integration, as well as main theorems of measure and integral.

COURSE DESCRIPTION AND SYLLABUS:
1. introduction. Semirings and rings of sets. Sigma - ring of sets.
2. Finitely and countably additive functions. Measures on rings.
3. Caratheodory construction.
4. Lebesgue measure. Regularity. Measures on reals.
5. Measurable functions. Integral of a simple function.
6. Integral of a nonnegative function. Monotone convergence theorem.
7. Integrable functions. Dominated convergence theorem.
8. Lebesgue Lp - spaces.
9. Hahn and Jordan decomposition of a measure. Absolute continuity. Singularity.
10. Radon - Nikodym theorem. Lebesgue decomposition.
11. Convergence types. Egorov theorem.
12. Product measures. Fubini - Tonelli theorem.
13. Riesz representation theorem of positive linear functionals.
Literature:
1. D. L. Cohn: Measure Theory
2. S. Mardešić: Matematička analiza 2. Integral i mjera
 1. semester Analiza - Regular study - Mathematics and Computer Science Education 2. semester Not active Analiza - Regular study - Mathematics and Computer Science Education
Consultations schedule:

### Content

Link to the course web page: https://web.math.pmf.unizg.hr/nastava/mii/