4 semester






2, Mathematics (economic activity)


Semester of training







Antonevich Anatoly Borisovich


Objectives of studying the discipline

To create a knowledge base and skills for students in the field of probability theory and mathematical statistics

acquaint students with the basic principles of probability theory and examples of their applications

further formation of students’ skills in abstract mathematical thinking and the ability to apply it in specific tasks, enhancing their mathematical culture

As a result of the study, the student should be able to:

be able to:

use the basic laws of random phenomena;

apply methods of probability theory and mathematical statistics in other sciences;



Algebra and number theory,

Discrete Math,

Analytic geometry,

Mathematical analysis,

Differential equations,

Functional Analysis


Contents of the discipline

Topic 1. Lebesgue measure. Systems of sets. Rings of sets, semirings, algebras, sigma-rings and sigma algebras, Borel sets. General concept of measure. Sigma-additive measure. Continuation of the Lebesgue measure. An exterior measure, measurable sets, sets of measure zero, the main theorem of measure theory. Lebesgue measure on a line. Lebesgue-Stieltjes measure.

Topic 2. The Lebesgue integral. Measurable functions, simple functions, integral of a simple function, integral of a measurable function, simplest properties of the Lebesgue integral, passage to the limit under the Lebesgue integral sign. The product of measures, The Fubini theorem.

Topic 3. Metric spaces. Definition and basic examples of functional metric spaces. Topology of metric spaces. Complete metric spaces. Replenishment of metric spaces. Compact metric spaces.

Topic 4. Continuous, uniformly continuous and Lipschitz mappings. Definitions and properties. The continuation theorem. Application of the principle of contracting mappings to integral equations.



Main literature:


1. Antonevich AB, Radyno Ya.V. Functional analysis and integral equations. 2 nd ed., Revised. and additional. Minsk, Publishing house of the Belarusian State University, 2006.

2. Antonevich AB, Mazel M.Kh., Radyno Ya.V. Functional analysis and integral equations. Tutorial. Minsk, Publishing house of the Belarusian State University, 2011.

3. Kolmogorov AN, Fomin S.V. Elements of the theory of functions and functional analysis. M., Fizmatlit, 2004.

4. Lyusternik LA, Sobolev VI Short course of functional analysis. M., Higher School, 1982.

5. Trenogin V.A. Functional analysis. M., Fizmatlit, 2002.


Additional literature:


1. Berezanskii Yu.M., Us G.Yu., Sheftel Z.G. Functional analysis. Lecture course. Kiev, High school, 1990.

2. Kantorovich LV, Akilov GP Functional analysis. St. Petersburg, Nevsky Dialect, BHV-Petersburg, 2002.

3. Kirillov AA, Gvishiani AD Theorems and problems of functional analysis. M., Science, 1979.

4. Antonevich AB, Knyazev PN, Radyno Ya.V. Tasks and exercises on functional analysis. Minsk, Higher School, 1978.


Teaching Methods

interactive methods of teaching (working in small groups (team), problem training) are organized taking into account the inclusion in the learning process of all students of the group. Joint activity means that each student makes his own individual contribution, in the course of the work there is an exchange of knowledge, ideas, methods of activity. Organized individual, steam and group work. Interactive methods are based on the principles of interaction, activity of trainees, reliance on group experience, mandatory feedback





Conditions (requirements), current control

– test;

– colloquium

The score on the exam is set taking into account: The score on the exam is set taking into account: 30% – work in laboratory and practical classes, 70% – oral examination.


Appraisal Form