# Measure Theory

Vrije Universiteit Amsterdam

## Course Description

• ### Course Name

Measure Theory

• ### Host University

Vrije Universiteit Amsterdam

• ### Location

Amsterdam, The Netherlands

Mathematics

• ### Language Level

Taught In English

• ### Course Level Recommendations

Upper

ISA offers course level recommendations in an effort to facilitate the determination of course levels by credential evaluators.We advice each institution to have their own credentials evaluator make the final decision regrading course levels.

### Hours & Credits

• ECTS Credits

6
• Recommended U.S. Semester Credits
3
• Recommended U.S. Quarter Units
4
• ### Overview

COURSE OBJECTIVE

Basics of measure theory and the Lebesgue integral

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 additivity of the measure. This leads to the notion of a sigma-algebra.

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 when working on the real line, but also allows one to work in an abstract setting. This is of crucial importance for the development of (for example) functional analysis and probability theory.

TEACHING METHODS
Classical classes with exercise classes.

TYPE OF ASSESSMENT
Written final exam, with 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.

ENTRY REQUIREMENTS
Single Variable Calculus, Multivariable Calculus and Mathematical Analysis (or equivalent)

### Course Disclaimer

Courses and course hours of instruction are subject to change.

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