Measure Theory (MAT351) | GSÜ - ECTS Information Guide

Content

Objective

To introduce the fondamentals of measure theory, with its applications to integration theory and probability theory.

Course Content

Measures and sigma-algebras,
Borel and Lebesgue measurable sets,
Measurable functions,
Lebesgue measure on R^n,
Lebesgue integral,
Convergence theorems: Monotone Convergence Theorem (MCT), Fatou's Lemma, Dominated Convergence Theorem (DCT)

Teaching and Learning Methods

Lectures
Exercices
Discussions

References

Mesure, Intégration, Eléments d'Analyse Fonctionelle, lecture notes of Petru Mironescu, Université Claude Bernard Lyon 1:
https://math.univ-lyon1.fr/~mironescu/resources/complet_mesure_integration.pdf

Weekly Contents

Theory Topics
Week	Weekly Contents
1	Review of Riemann Integral, Motivation for the need of a more general theory
2	sigma-algebras, Borel sets
3	Measurable functions
4	Measures
5	Negligable sets, Lebesgue measure
6	Constrution of Lebesgue measure
7	Integral of a measurable function
8	Midterm
9	Integral of a measurable function, Monotone Convergence Theorem
10	Integration
11	Fatou's Lemma, Dominated Convergence Theorem
12	Product measure
13	Fubini-Tonelli Theorem
14	Fubini-Tonelli Theorem

Assessment System

Contribution to Overall Grade
Activities	Number	Contribution
Contribution of in-term studies to overall grade	3	60
Contribution of final exam to overall grade	1	40
Total	4	100

In-Term Studies
Activities	Number	Contribution
Assignments	2	30
Presentation	0	0
Midterm Examinations (including preparation)	1	30
Project	0	0
Laboratory	0	0
Other Applications	0	0
Quiz	0	0
Term Paper/ Project	0	0
Portfolio Study	0	0
Reports	0	0
Learning Diary	0	0
Thesis/ Project	0	0
Seminar	0	0
Other	0	0
Make-up	0	0
Total	3	60

Relation of Proficiency

No	Program Learning Outcomes	Contribution
No	Program Learning Outcomes	1	2	3	4	5
1	understands principles of deductive reasoning; has experience to verify well-foundedness and exactness of mathematical statements in systematic ways;					X
2	can properly state and use concepts and results of major mathematical interest;					X
3	masters current computational techniques and algorithms; has a good ability in their use; can identify relevant tools, among those one has learned, suitable to solve a problem and is able to judge whether or not one is in possession of these tools;				X
4	is able to express one’s mathematical ideas in an organised way both in written and oral forms;				X
5	understands relations connecting substantial concepts and results; can switch from one viewpoint to another on mathematical objects (pictures, formulae, precise statements, heuristic trials, list of examples,...);					X
6	has followed individually a guided learning strategy; has pursued steps toward the resolution of unfamiliar problems;
7	has a theoretical and practical knowledge in computer science well adapted for learning a programming language;
8	has investigated the relevance of modeling and using mathematical tools in natural sciences and in the professional life; is conscious about historical development of mathematical notions;	X
9	has followed introduction to some mathematical or non-mathematical disciplines after one’s proper choice; had experience to learn selected subjects according to one’s proper arrangement;	X
10	masters French language as well as other foreign languages, to a level sufficient to study or work abroad.			X

ECTS

Activities	Number	Period	Total Workload
Class Hours	14	3	42
Working Hours out of Class	14	3	42
Assignments	4	5	20
Presentation	0	0	0
Midterm Examinations (including preparation)	1	6	6
Project	0	0	0
Laboratory	0	0	0
Other Applications	0	0	0
Final Examinations (including preparation)	1	10	10
Quiz	0	0	0
Term Paper/ Project	0	0	0
Portfolio Study	0	0	0
Reports	0	0	0
Learning Diary	0	0	0
Thesis/ Project	0	0	0
Seminar	0	0	0
Other	0	0	0
Make-up	0	0	0
Yıl Sonu	0	0	0
Hazırlık Yıl Sonu	0	0	0
Hazırlık Bütünleme	0	0	0
Total Workload			120
Total Workload / 25			4.80
Credits ECTS			5