# MATH5825 Measure, Integration and Probability

MATH5825 is a Honours and Postgraduate Coursework Mathematics course. See the course overview below.

Units of credit: 6

Prerequisites:

Cycle of offering: Yearly in Semester 2.

Graduate attributes: The course will enhance your research, inquiry and analytical thinking abilities.

More information: This recent course handout (pdf) contains information about course objectives, assessment, course materials and the syllabus.

The Online Handbook entry contains information about the course. (The timetable is only up-to-date if the course is being offered this year.)

If you are currently enrolled in MATH5825, you can log into UNSW Moodle for this course.

#### Course Overview

Measure Theory provides one of the key building blocks of the modern theory of Analysis, Probability Theory, and Ergodic Theory and has important applications in the theory of differential equations, Harmonic Analysis, Theoretical Physics and Mathematical Finance.

In this course we will develop a proper understanding of measurable functions, measures and the Lebesgue integral. Given these concepts we will consider various concepts of convergence of measurable functions and the convergence of the corresponding integrals, changes of measures and spaces of integrable functions. A special attention will be paid to applications of Measure Theory in the Probability Theory. First we will develop a proper understanding of probability spaces for random variables and their finite and infinite sequences. Using these concepts we will discuss Strong Laws of Large Numbers and their applications. Changes of measures and the Radon-Nikodym Theorem will be applied to introduce a general definition of conditional expectation and to study their properties. Then we will apply this machinery to study Gaussian systems and we will introduce the so-called chaotic decompositions which provide an important tool for the Malliavin Calculus, Finance and Physics. Finally, we will introduce the weak convergence of measure, characteristic functions. We will use this theory to derive the Central Limit Theorem and we will discuss some of its applications.