Measure and Integral Theory 360-MS2-1TM

obligatory courses

(in Polish) zdalnie
(in Polish) w sali
Blended learning

Objectives and objectives of the course: After completing the course, the student should: understand the difference, and more precisely the advantage of the Lebesgue integral over the Riemann integral; see and distinguish metric structures, including structures on family sets; know and be able to use the basic theorems about crossing the boundary into the integral and the Radon-Nikodym theorem; as well as understand the basic literature on the subject, including the ability to independently deepen and expand knowledge in a given field.

Understands differences and the advantage of the Lebesgue integral over the Riemann integral; knows the basic properties of the Lebesgue integral. He knows the basic theorems about crossing the boundary under the sign of the integral and the Radon-Nikodym theorem; understand the Radon-Nikodym derivative. He can calculate integrals of simple functions with respect to abstract measures. He can distinguish between metric structures, including structures on families of sets. He can apply the basic theorems about crossing the limit into the integral sign.

KA7_WG03, KA7_UW02, KA7_UW07

General form of assessment: exam (oral or oral and written) and tests

The exam topics include:

1. σ-algebras of sets: definition, properties, examples, σ-algebras generated by families of sets, σ-algebra of Borel sets on Rn - different types of generators.

2. Measures: properties, continuity of measure vs. σ-additivity, examples of measures.

3. Uniqueness of measure: λ-systems, the Lemma on λ and π-systems, and Dynkin's uniqueness theorem.

4. Corollaries of the Uniqueness Theorem: uniqueness of the Lebesgue measure, invariance of measures under shifts on Rn.

5. Caratheodori's theorem on the extension of a measure from a semiring to a σ-algebra of sets: semiring, outer measure, extension from a semiring to a ring, measurable sets (satisfying the Caratheodori condition).

6. Existence of the Lebesgue-Stieltjes measure (σ-additivity).

7. Measurable mappings: characterizations, properties, examples.

8. The image of a measure under a measurable mapping: definition, examples, the image of the Lebesgue measure under invertible linear mappings.

9. Real measurable functions and simple functions: definitions, examples, properties, approximation of measurable functions by simple functions.

10. Definition and properties of an integral: formula for simple functions, nonnegative functions, and general definition. Properties of the integral and examples.

11. Radon-Nikodym theorem: absolutely continuous measures, examples.

12. Change of variables in an integral: General theorem, corollaries, and examples (absolute continuity of the Lebesgue-Stieltjes measure, continuous distributions in probability).

13. Theorems on transitions with a limit under an integral: Levi's theorem (on monotone convergence), Lebesgue's theorem (on majorized convergence), and Fatou's lemma, with examples.

14. Product measures and iterated integrals: construction and existence of a product measure, Tonelli's theorem, and Fubini's theorem.

In accordance with § 9 of Order No. 31 of the Rector of the University of Białystok of April 11, 2025, the following are prohibited for the limited use of AI systems in written activities by persons developing in the interior: 1) making AI systems available to persons prohibited under § 4 point 3, 2) releasing AI systems to a different extent or in another manner presented by the wide use, 3) using documentation of the use of AI, 4) automatic execution of a task in or part by AI systems, 5) disclosing access to the AI ​​system as a source of bibliographic information, 6) not originating from the AI ​​system as legal sources.

P. Halmos, Measure theory, van Nostrand, Princeton, 1956.

S. Łojasiewicz, Wstęp do teorii funkcji rzeczywistych, PWN, 1987

W. Rudin, Analiza rzeczywista i zespolona, PWN, Warszawa, 2009.

A. Birkholc Analiza matematyczna. Funkcje wielu zmiennych, PWN 1986.

Additional information (registration calendar, class conductors, localization and schedules of classes), might be available in the USOSweb system:

• Description of 360-MS2-1TM in USOSweb