This module provides an introduction to the basic ideas and concepts of measure theory and and the Lebesgue integral, which generalised the Riemann integral. Measure theory is indispensable for almost all studies in higher mathematics and especially for all subjects based on Hilbert spaces and linear operators (e.g., Quantum mechanics, spectral theory and quantum field theory, representation theory, operator algebras), for many areas of advanced probability, partial differential equations (solutions in Sobolev spaces), calculus of variations (generalised manifolds, geometric measure theory) and for a rigorous understanding of fractal objects. (Hausdorff dimension, Hausdorff measure.)

• Know and understand the concept of a sigma-algebra and a measure;
• Know and understand the concept of the Lebesgue measure;
• Know and understand the concept of almost everywhere prevailing properties;
• Understand the Radon-Nikodym theorem;
• Understand the relation between convergence of Lebesgue integrals and pointwise convergence of functions;
• Know and understand products measures and Fubini's theorem;

30 - 50 minute lectures

Some handouts will be provided in hard copy or via Learning Central, but students will be expected to take notes of lectures.

Students are also expected to undertake at least 120 hours private study including preparation of worked solutions for problem classes.

The ability to understand courses which require measure theory as a prerequisite, e.g., functional analysis.

Problem solving

Mathematical abstraction

Transferable Skills:
Appreciation and understanding of measure theory; ability to understand mathematical concepts based on integration theory.

Problem solving

Mathematical abstraction

Formative assessment is carried out in the problem classes.  Feedback to students on their solutions and their progress towards learning outcomes is provided during these classes.  

The summative assessment is the written examination at the end of the module.  This gives students the opportunity to demonstrate their overall achievement of learning outcomes.  It also allows them to give evidence of the higher levels of knowledge and understanding required for above average marks.

The examination paper has a choice of four from five equally weighted questions.

• Sigma-algebras and measures
• Lebesgue outer measure, Lebesgue measurable sets and Lebesgue measure.
  Non Lebesgue measurable sets and the axiom of choice.
• Hausdorff measure and Hausdorff dimension of fractal objects
• Measurable functions and image measure
• Integrability (in the sense of Lebesgue) and almost everywhere prevailing properties
• Integral convergence theorems
• Radon-Nikodym theorem
• Spaces of integrable functions
• Dynkin (d-) systems and the monotone class theorem with applications
• Product measures, product Sigma-algebras and Fubini's theorem

Bauer, H., Measure and Integration Theory, Walter de Gruyter

Halmos, P., Measure Theory, Springer

Lieb, E., and Loss, M., Analysis, American Mathematical Society, (2001)

Not applicable.