### General Information

Course Code | X_401028 |
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Credits | 6 EC |

Period | P1+2 |

Course Level | 300 |

Language of Tuition | English |

Faculty | Faculty of Science |

Course Coordinator | prof. dr. R.W.J. Meester |

Examiner | prof. dr. R.W.J. Meester |

Teaching Staff |
prof. dr. R.W.J. Meester |

### Practical Information

You need to register for this course yourself

Last-minute registration is available for this course.

Teaching Methods | Seminar, Lecture |
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Target audiences

This course is also available as:

### Course Objective

After this course...1. ... the student will know and understand the basic

concepts of measure theory and the theory of Lebesgue integration.

2. ... the student will understand the main proof techniques in the

field, and he will also be able to apply the theory abstractly and

concretely.

3. ... the student will be able to write elementary proofs himself, as

well as more advanced proofs under guidance.

4. ... the student is able to use measure theory and integration in

Riemann integration and calculus.

5. ... the student is able to work with Lebesgue measure and to exploit

its special properties.

### Course Content

We motivate and introduce the notion of a measure, that is, a way toassign a size to as many subsets as possible in an abstract space. It

turns out that it is in general not possible to measure all sets, at

least if one insists on countable additivity of the measure. This leads

to the notion of a sigma-algebra. We show how one can obtain a unique

measure on a sigma algebra once certain basic properties are imposed.

Once we have defined measure, we can introduce and discuss so called

measurable functions which, roughly speaking, form the class of

functions which we will be able to integrate. We then introduce and

study integration of these measurable functions with respect to a

measure. We discuss (among other things) the monotone and dominated

convergence theorems concerning the interchangeability of limit and

integral, the substitution rule, absolute continuity and the relation of

this new integral to the Riemann integral. We also discuss

multi-dimensional Lebesgue measures, product measures, and Fubini’s

theorem.

The theory leads to a new perspective on integration of functions, which

is not only more general than the Riemann setting when working on the

real line, but also allows one to integrate in an abstract setting. This

is of crucial importance for the development of functional analysis and

probability theory.

### Teaching Methods

Lectures and exercise classes.### Method of Assessment

Written final exam, and a written midterm exam after 7 weeks. The finalexam will be 50% of the final grade, and the midterm exam will be 40%.

The remaining 10% will be homework, but the homework only counts if the

weighted average of the two exams is at least 5,50. The homework does

not count towards the resit exam.

### Entry Requirements

Single Variable Calculus, Multivariable Calculus and MathematicalAnalysis (or equivalent).

### Literature

Rene L. Schilling: Measures, Integrals and Martingales, CambridgeUniversity Press.