The aim of studying and bibliography

The main aim of study: To familiarize students with the Lebesgue integral and its basic properties, and with profound theorems concerning differentiability almost everywhere and absolute continuity.

The general information about the module: The module is implemented in the second 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: Knowledge of the material realized in the framework of the module Real functions I.

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.: exterior measure, Caratheodory's condition, Lebesgue measure, simple function, characteristic function of a set, measurable function, almost everywhere, in measure and almost uniform convergent sequences of measurable functions, Lebesgue integral, relation between Riemann integral and Lebesgue integral	lecture, exercises	test, exam	K_W01+ K_W02+ K_W03+ K_W04++ K_W05++ K_W07+ K_U15+ K_K01+ K_K02+ K_K04+ K_K07+	X2A_W02+ X2A_W03+ X2A_U06+ X2A_U07+ X2A_U08+ X2A_U09+ X2A_K01+ X2A_K02+ X2A_K03+ X2A_K04+ X2A_K06+
02	knows how to check if a given function is exterior measure	lecture, exercises	test, exam	K_W01+ K_W02+ K_W03+ K_U01+ K_U02+ K_U03+ K_U04+ K_U05+ K_U07+ K_U08+ K_U13+	X2A_W03+ X2A_U01+ X2A_U02+ X2A_U03+ X2A_U05+
03	knows how to calculate or estimate Lebesgue measure of a set contained in R or R2	lecture, exercises	test, exam	K_W01+ K_W02+ K_W03+ K_U02+ K_U05+ K_U07++ K_U08+ K_U09+ K_U13+ K_U14+	X2A_W03+ X2A_U01+ X2A_U02+ X2A_U03+ X2A_U05+
04	knows how to check if the function is measurable in the Lebesgue sense	lecture, exercises	test, exam	K_W01+ K_W02+ K_W03+ K_U02+ K_U07++ K_U13+	X2A_W03+ X2A_U01+ X2A_U02+ X2A_U03+ X2A_U05+
05	knows how to verify if the given sequence of measurable functions is convergent almost everywhere, if it is convergent in measure and whether it is almost uniform convergent	lecture, exercises	test, exam	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+	X2A_W03+ X2A_U01+ X2A_U02+ X2A_U03+ X2A_U05+
06	knows how to calculate Lebesgue integral of simple function and knows how to calculate Lebesgue integral useing the properties of Lebesgue integral	lecture, exercises	test, exam	K_W01+ K_W02+ K_W03+ K_U02+ K_U07++	X2A_W03+ X2A_U01+ X2A_U03+ X2A_U05+

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
2	TK01	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.	W01-W02, C01-C08	MEK01 MEK02 MEK03
2	TK02	Definition and properties of measurable functions. Luzin theorem and Frechet theorem. Baire functions. Vitali theorem.	W03-W08, C09-C12	MEK01 MEK04
2	TK03	Sequences of measurable functions. Convergence almost everywhere, convergence in measure, almost uniform convergence. Jegorov theorem and Riesz theorem.	W09-W14, C13-C18	MEK01 MEK04 MEK05
2	TK04	Integral of nonnegative function and its properties. Integral of functions with respect to a measure of any sign. Examples and counter-examples related to integral with respect to measure.	W15-W20, C19-C24	MEK01 MEK04 MEK06
2	TK05	Lebesgue integral. Examples and counter-examples related to Lebesgue integral. Absolute continuity of integral. Properties of integral with variable upper limit of integration. Fatou lemma. Lebesgue theorem on monotone and dominated convergence. Vitali theorem. Relation between Riemann integral and Lebesgue integral.	W21-W30, C25-C30	MEK01 MEK04 MEK06

The student's effort

The type of classes	The work before classes	The participation in classes	The work after classes
Lecture (sem. 2)		contact hours: 30.00 hours/sem.	complementing/reading through notes: 10.00 hours/sem.
Class (sem. 2)	The preparation for a Class: 30.00 hours/sem. The preparation for a test: 10.00 hours/sem.	contact hours: 30.00 hours/sem.
Advice (sem. 2)	The preparation for Advice: 4.00 hours/sem.	The participation in Advice: 4.00 hours/sem.
Exam (sem. 2)	The preparation for an Exam: 10.00 hours/sem.	Others: 2.00 hours/sem.

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 the result of the written or oral exam. There is a possibility of exemption from the exam based on a credit for the exercises.
Class	A credit for the exercises is based on the results of tests and oral answers.
The final grade	The final grade is the credit for the exam and the credit of the exercises (the arithmetic mean).

Sample problems

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

Realized during classes/laboratories/projects
przykładowe zadania funkcje rz II.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