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## Measure Theory

2019-2020

### 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 to
assign 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 final
exam 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 Mathematical
Analysis (or equivalent).

### Literature

Rene L. Schilling: Measures, Integrals and Martingales, Cambridge
University Press.

3MAT, 3EOR

### General Information

Course Code X_401028 6 EC P1+2 300 English Faculty of Science prof. dr. R.W.J. Meester prof. dr. R.W.J. Meester 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
Target audiences

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