 systems of sets, Caratheodory theorem,
 ddimensional Lebesgue measure,
 measurable functions, integral w.r.t. a measure, convergence theorems,
 L^{p} spaces,
 product measures, Fubini's theorem,
 transformation theorem,
 theorem of RadonNikodym.
Module MAT122M3
Measure Theory and Differential Equations (M, 9.0 LP)
Module Identification
Module Number  Module Name  CP (Effort) 

MAT122M3  Measure Theory and Differential Equations  9.0 CP (270 h) 
Basedata
CP, Effort  9.0 CP = 270 h 

Position of the semester  1 Sem. in SuSe 
Level  [3] Bachelor (Core) 
Language  [DE] German 
Module Manager  
Lecturers 
Lecturers of the department Mathematics

Area of study  [MATGRU] Mathematics (B.Sc. year 1 and 2) 
Reference course of study  [MAT82.276SG] B.Sc. Business Mathematics 
LivecycleState  [NORM] Active 
Courses
Type/SWS  Course Number  Title  Choice in ModulePart  PresenceTime / SelfStudy  SL  SL is required for exa.  PL  CP  Sem.  

2V+1U  MAT1228K3  Measure and Integration Theory
 P  42 h  93 h 
USchein
   PL1  4.5  SuSe 
2V+1U  MAT1225K3  Introduction to Ordinary Differential Equations
 P  42 h  93 h 
USchein
   PL1  4.5  SuSe 
 About [MAT1228K3]: Title: "Measure and Integration Theory"; PresenceTime: 42 h; SelfStudy: 93 h
 About [MAT1228K3]: The study achievement [USchein] proof of successful participation in the exercise classes (ungraded) must be obtained.
 About [MAT1225K3]: Title: "Introduction to Ordinary Differential Equations"; PresenceTime: 42 h; SelfStudy: 93 h
 About [MAT1225K3]: The study achievement [USchein] proof of successful participation in the exercise classes (ungraded) must be obtained.
Examination achievement PL1
 Form of examination: oral examination (2030 Min.)
 Examination Frequency: each semester
 Examination number: 82026 ("Module Exam Measure Theory and Differential Equations")
Evaluation of grades
The grade of the module examination is also the module grade.
Contents
 firstorder differential equations: autonomous firstorder differential equations, variation of constants, explicitly solvable cases, initial value problems,
 existence and uniqueness: functionalanalytical foundations, Banach fixedpoint theorem, PicardLindelöf theorem, the continuability of solutions, Peano's existence theorem,
 qualitative behaviour: Gronwall's lemma, continuous dependency on data, upper and lower functions,
 linear differential equations: homogeneous linear systems, matrix exponential function, variation of constants, nthorder differential equations,
 stability: dynamical systems, phase space, Hamiltonian systems, asymptotic behaviour, stability theory according to Lyapunov.
Competencies / intended learning achievements
They know and understand the basic concepts, constructions, results and methods of proof of measure and integration theory. They are able to comprehend the proofs and independently prove or disprove statements. They are able to correctly handle measuretheoretical constructions as well as the Lebesgue integral in a general measuretheoretical context and with regard to its convergence properties. This knowledge is a prerequisite for all advanced courses in the fields of stochastics and functional analysis.
In addition, the students know and understand the basic concepts, statements and methods of the theory of ordinary differential equations, and they master their analytical treatment. By combining results from analysis and linear algebra, they are able to investigate advanced problems and to work on smaller application problems from science and technology using mathematical methods.
In the exercise classes they have acquired a confident, precise and independent handling of the terms, propositions and methods from the lectures.
Literature
 J. Elstrodt: Maß und Integrationstheorie,
 H. Bauer: Maß und Integrationstheorie.
 V.I. Arnold: Gewöhnliche Differentialgleichungen,
 L. Grüne, O. Junge: Gewöhnliche Differentialgleichungen,
 H. Heuser: Gewöhnliche Differentialgleichungen,
 J.W. Prüss, M. Wilke: Gewöhnliche Differentialgleichungen und dynamische Systeme,
 W. Walter: Gewöhnliche Differentialgleichungen,
 G. Teschl: Ordinary Differential Equations and Dynamic Systems.
Registration
Requirements for attendance (informal)
Modules:
Requirements for attendance (formal)
References to Module / Module Number [MAT122M3]
Course of Study  Section  Choice/Obligation 

[MAT82.276SG] B.Sc. Business Mathematics  General Mathematics  [P] Compulsory 