1. komponenta

Lecture type	Total
Lectures	30
Exercises	30

* Load is given in academic hour (1 academic hour = 45 minutes)

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.

Analiza - Regular study - Mathematics Education

Analiza - Regular study - Mathematics Education