Measure Theory module (MA51007)

About the module

This module, aimed at the Level 5 student, covers the fundamentals of Measure Theory including Lebesgue Measure and Integration and Lebesgue Spaces as well as their pivotal implications in the modern analysis of partial differential equation and probability theory. This module may optionally be taken in combination with other modules at this level by Level 5 Students on the MMath in Mathematics or MSci in Mathematics and Physics degrees. If you have questions about this module or the possible combinations, please contact your Advisor of Studies.

Prerequisites

Students taking this module must usually have achieved a pass mark in the module MA32001 or equivalent.

Indicative Content

• Construction of a general measure space

  Definition and properties of a σ-algebra. The concept of measures as a non-negative σ-additive real valued set functions defined on a σ-algebra. Measurable sets, σ-finite measurable sets, the notion of algebra of generators of a σ- algebra of parts. Construction of the Lebesgue Measure. Charatheodory Extension Theorem: characterising the unique extension of a finitely additive non-negative real valued function on an algebra of generators to a measure on the generated σ-algebra.

• Measurable functions

  Definition. Approximation by measurable simple/step functions as well as by continuous functions.

• Construction of the Lebesgue Integral

  Construction of the integral for measurable indicators (characteristic) functions as well as for measurable simple/step functions. Construction of the integral for a general measurable function. Properties of Lebesgue Integrable functions (additivity, multiplication by scalars, positivity, monotonicity). Definition and basic vectorial properties of the space of Lebesgue Integrable Functions L¹.

• Limit theorems (concerning point-wise convergence, almost everywhere convergence, and convergence in measure)

  Fatou Lemma. Monotone Convergence Theorem. Dominated Convergence Theorem.

• Product Measures and the Fubini Theorem

• Absolutely continuous measures and the Lebesgue-Radon-Nikodym Theorem

• Definition and properties of general Lebesgue Spaces L^p and their embedding relations

If time permits then the following will also be covered:

• The Sobolev Spaces H¹ and their implications for the analysis of elliptic equations partial differential equations

• Connection with probability theory, including random variables, cumulative distribution functions and martingales

Delivery and Assessment

The module is delivered in the form of lectures and seminars and assessed via an exam (60%) and coursework (40%).

Credit Rating

This module is a Scottish Higher Education Level 5 or SCQF level 11 module and is rated as 15 SCOTCAT credits or 7.5 ECTS credits.