## MAS350 Measure and Probability

 Semester 2, 2021/22 10 Credits Lecturer: Dr Nic Freeman Home page Timetable Reading List Aims Outcomes Teaching Methods Assessment

Measure theory is that branch of mathematics which evolves from the idea of "weighing" a set by attaching a non-negative number to it which signifies its worth. This generalises the usual physical ideas of length, area and mass as well as probability. It turns out (as we will see in the course) that these ideas are vital for developing the modern theory of integration.

The module will give students an additional opportunity to develop skills in modern analysis as well as providing a rigorous foundation for probability theory. In particular it would form a useful precursor or companion course to the Level 4 courses MAS436 (Functional Analysis) and MAS452 (Stochastic Processes and Finance), the latter of which is fundamentally dependent on measure theoretic ideas.

Prerequisites: MAS211 (Advanced Calculus and Linear Algebra); MAS221 (Analysis)
Not with: MAS451 (Measure and Probability)
No other modules have this module as a prerequisite.

## Outline syllabus

• The scope of measure theory,
• σ-algebras,
• Properties of measures,
• Measurable functions,
• The Lebesgue integral,
• Interchange of limit and integral,
• Probability from a measure theoretic viewpoint,
• Characteristic functions,
• The central limit theorem.

## Aims

• give a more rigorous introduction to the theory of measure.
• develop the ideas of Lebesgue integration and its properties.
• recall the concepts of probability theory and consider them from a measure theoretic point of view.
• prove the Central Limit Theorem using these methods.

## Learning outcomes

• understand why a more sophisticated theory of integration and measure is needed;
• show that certain functions are measurable;
• construct the Lebesgue integral;
• understand properties of the Lebesgue integral;
• develop probabilistic concepts (random variables, expectation and limits) within the framework of measure theory.

## Teaching methods

Lectures and problem solving.

20 lectures, no tutorials

## Assessment

One 2 hours 30 minutes exam