Modules

Prerequisites

Contents

• Algebras of sets and measures: σ-algebras and their generators, outer measure, pre-measures, measures, Caratheodory condition, extension of pre-measures to measures, Lebesgue measure on R, uniqueness theorems for measures • Measurable functions and integration: measurable maps and image measures, integration of measurable functions, theorems of Beppo-Levi, Fatou and Lebesgue, measures with densities, theorem of Radon-Nikodym • Product measures: products of measurable spaces, product measures, Fubini's theorem, Lebesgue measure in n-dimensional space, transformation rule • Absolute continuity and Lebesgue's differentiation theorem • Hausdorff measure

Learning outcomes

After successfully completing the module, students know the basic concepts of measure and integration theory and are enabled to deal with them confidently. They possess an enhanced capacity for abstraction.

Assessment

Prerequisite: • Successful completion of homework Final module examination: • Written examination, 90 min. OR • Oral examination, 30 min. It will be announced in the first course whether the examination will organized in written or oral form.

Remarks

• Study programme Mathematics M.Sc.: Compulsory elective module in the complex „Stochastics“ • Study programme Mathematik B.Sc.: Compulsory elective module in the complex „Vertiefung“ (to a limited extend) • Study programme Wirtschaftsmathematik B.Sc.: Compulsory elective module in the complex „Vertiefung“ (to a limited extend)