### MA4007: Measure Theory

 School Cardiff School of Mathematics Department Code MATHS Module Code MA4007 External Subject Code G100 Number of Credits 20 Level L7 Language of Delivery English Module Leader Mr Frank Rosler Semester Autumn Semester Academic Year 2019/0

#### Outline Description of Module

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), calculus of variations (generalised manifolds, geometric measure theory) and for a rigorous understanding of fractal objects. (Hausdorff dimension, Hausdorff measure.)

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

#### On completion of the module a student should be able to

• 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;

#### How the module will be delivered

30 fifty-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.

#### Skills that will be practised and developed

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

#### How the module will be assessed

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.

#### Assessment Breakdown

Type % Title Duration(hrs)
Examination - Autumn Semester 100 Measure Theory 3

#### Syllabus content

• 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