This module provides an introduction to the basic ideas and concepts of measure theory 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), and in calculus of variations (generalised manifolds, geometric measure theory).

Recommended Modules: MA2006 Real Analysis, MA3005 Functional and Fourier Analysis

• 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 relation between convergence of Lebesgue integrals and pointwise convergence of functions;
• Know and understand products measures and Fubini's theorem;

Modules will be delivered through blended learning. You will be guided through learning activities appropriate to your module, which may include:

• On-line resources that you work through at your own pace (e.g. videos, web resources, e-books, quizzes),
• On-line interactive sessions to work with other students and staff (e.g. discussions, live streaming of presentations, live-coding, group meetings)
• Face to face small group sessions (e.g. tutorials, exercise classes, feedback sessions)

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

• Sigma-algebras and measures
• Lebesgue outer measure, Lebesgue measurable sets and Lebesgue measure.
• Non Lebesgue measurable sets and the axiom of choice.
• Measurable functions
• Integrability (in the sense of Lebesgue) and almost everywhere prevailing properties
• Integral convergence theorems
• The Riesz-Fisher theorem
• Product measures, product Sigma-algebras and Fubini's theorem

An interactive version of this list is available at https://whelf-cardiff.alma.exlibrisgroup.com/leganto/public/44WHELF_CAR/lists/7729805940002420?auth=SAML