Measure Theory

Vrije Universiteit Amsterdam

Course Description

  • Course Name

    Measure Theory

  • Host University

    Vrije Universiteit Amsterdam

  • Location

    Amsterdam, The Netherlands

  • Area of Study

    Mathematics

  • Language Level

    Taught In English

  • Course Level Recommendations

    Upper

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    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
    Basics of calculus.

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