The aim of studying and bibliography

The main aim of study: To familiarize students with the fundamentals of measure theory with particular emphasis on Lebesgue measure.

The general information about the module: The module is implemented in the first semester in the form of lectures (30 hours) and exercises (30 hours).

Bibliography required to complete the module

Bibliography used during lectures

1	E. DiBenedetto	Real Analysis	Birkhäuser, Springer, New York.	2016
2	S. Łojasiewicz	Wstęp do teorii funkcji rzeczywistych	PWN, Warszawa.	1973
3	W. Rudin	Analiza rzeczywista i zespolona	PWN, Warszawa.	1986
4	R. Sikorski	Funkcje rzeczywiste, tom I	PWN, Warszawa.	1958

Bibliography used during classes/laboratories/others

1	W. Rudin	Analiza rzeczywista i zespolona	PWN, Warszawa.	1986
2	R. Sikorski	Funkcje rzeczywiste, tom I	PWN, Warszawa.	1958

Basic requirements in category knowledge/skills/social competences

Formal requirements: Basic knowledge of mathematical analysis of the first level studies: differential calculus of functions of one variable and functions of several variables, the Riemann’s integral, multiple integrals.

Basic requirements in category knowledge: A student has mathematical knowledge which allows him/her to understand the lectured material.

Basic requirements in category skills: Ability to use fundamental mathematical tools and the knowledge obtained during the first level studies.

Basic requirements in category social competences: A student is prepared to undertake substantiated mathematical operations in order to solve a task.

Module outcomes

MEK	The student who completed the module	Types of classes / teaching methods leading to achieving a given outcome of teaching	Methods of verifying every mentioned outcome of teaching	Relationships with KEK	Relationships with OEK
01	knows basic concepts and definitions given during the course of lectures i.e.: lower limit and upper limit of a sequence of sets, field of sets and sigma-field of sets, finitely additive measure and sigma additive measure, complete measure, Jordan measure	lecture, exercises	test	K_W01+ K_W02+ K_W03+ K_W04+++ K_W05++ K_W07+ K_K04+ K_K07+	X2A_W02+ X2A_W03+ X2A_K03+ X2A_K04+ X2A_K06+
02	knows how to perform fundamental operations on sets and is able to indicate lower limit and upper limit of a sequence of sets	lecture, exercises	test	K_W01+ K_W02+ K_W03+ K_U01++ K_U02++ K_U04+ K_U08+ K_K01+	X2A_W03+ X2A_U01+ X2A_U02+ X2A_U03+ X2A_U05+ X2A_U07+ X2A_K01+
03	knows how to check properties of family of sets	lecture, exercises	test	K_W01+ K_W02+ K_W03+ K_U01++ K_U02++ K_U04+ K_U07+ K_U08+ K_K01+	X2A_W03+ X2A_U01+ X2A_U02+ X2A_U03+ X2A_U05+ X2A_U07+ X2A_K01+
04	knows how to check if a given function is finitely additive measure and if it is sigma additive measure	lecture, exercises	test	K_W01+ K_W02+ K_W03+ K_U01++ K_U02++ K_U03++ K_U04+ K_U05+ K_U07+ K_U08+ K_U09+ K_U13+ K_U14+ K_K01+	X2A_W03+ X2A_U01+ X2A_U02+ X2A_U03+ X2A_U05+ X2A_U07+ X2A_K01+
05	knows how to calculate or estimate Jordana measure of a set contained in R or R^2	lecture, exercises	test	K_W01+ K_W02+ K_W03+ K_U01++ K_U02++ K_U03++ K_U04+ K_U07+ K_U15+ K_K01+ K_K02+	X2A_W03+ X2A_U01+ X2A_U02+ X2A_U03+ X2A_U05+ X2A_U06+ X2A_U07+ X2A_U08+ X2A_U09+ X2A_K01+ X2A_K02+

Attention: Depending on the epidemic situation, verification of the achieved learning outcomes specified in the study program, in particular credits and examinations at the end of specific classes, can be implemented remotely (real-time meetings).

The syllabus of the module

Sem.	TK	The content	realized in	MEK
1	TK01	Sets, operations on sets, family of sets, sequences of sets. Operations on family of sets. Lower limit and upper limit of a sequence of sets. Additive family, sigma additive family, difference family, multiplicative family, sigma multiplicative family, complementary family.	W01-W08, C01-C14	MEK01 MEK02 MEK03
1	TK02	Field of sets and sigma-field of sets. Sigma-field generated by any family of sets. Sigma-field of Borel sets. Finitely additive measure and its properties. Sigma additive measure. Space with a measure. Complete measure. Extension of measure to complete measure.	W09-W16, C15-C24	MEK01 MEK04
1	TK03	Definition and properties of Jordan measure. Cantor set and its Jordan measure. Measurable sets and non-measurable sets in the Jordan's sense.	W17-W22, C25-C30	MEK01 MEK05
1	TK04	Exterior measure. Caratheodory's condition. Metric exterior measure. Lebesgue measure. The structure of measurable sets in the Lebesgue's sense. Problems related to the determination of Lebesgue measure of sets and properties of measurable sets in the Lebesgue's sense. Examples of non-measurable sets in the Lebesgue's sense.	W23-W30	MEK01

The student's effort

The type of classes	The work before classes	The participation in classes	The work after classes
Lecture (sem. 1)		contact hours: 30.00 hours/sem.	complementing/reading through notes: 10.00 hours/sem.
Class (sem. 1)	The preparation for a Class: 30.00 hours/sem. The preparation for a test: 20.00 hours/sem.	contact hours: 30.00 hours/sem.
Advice (sem. 1)	The preparation for Advice: 4.00 hours/sem.	The participation in Advice: 4.00 hours/sem.
Credit (sem. 1)

The way of giving the component module grades and the final grade

The type of classes	The way of giving the final grade
Lecture	A credit for the lecture is based on attendance at the lectures.
Class	A credit for the exercises is based on the results of tests and oral answers.
The final grade	The final grade is a credit for the exercises.

Sample problems

Required during the exam/when receiving the credit
(-)

Realized during classes/laboratories/projects
przykładowe zadania funkcje rz I.pdf

Others
(-)

Can a student use any teaching aids during the exam/when receiving the credit : no

The contents of the module are associated with the research profile: no