Mathematics

Measure and Integration

Module code: G1070
Level 6
15 credits in spring teaching
Teaching method: Lecture
Assessment modes: Coursework, Unseen examination

In this module, you cover:

  • countably additive measures, sigma-algebras, Borel sets, measure spaces
  • outer measures and Caratheodory's construction of measures
  • construction and properties of Lebesgue measure in Euclidean spaces
  • measurable and integrable functions, Lebesgue integration theory on measure spaces, L^p spaces and their properties
  • convergence theorems: monotone convergence, dominated convergence, Fatou's lemma
  • application of limit theorems to continuity and differentiability of integrals depending on a parameter
  • properties of finite measure spaces and probability theory.

Module learning outcomes

  • A successful student should: know the concept and the properties of a sigma-additive measure;
  • understand how the concepts of integral and measure are related;
  • be able to state and apply central results of integration theory such as: monotone and dominated convergence theorems or theorems on parameter-dependent integrals;
  • know basic facts on L^p spaces.