Overview
MATH5825 is an honours and postgraduate mathematics course. See the course overview below.
Units of credit: 6
Cycle of offering: Term 3
Graduate attributes: The course will enhance your research, inquiry and analytical thinking abilities.
More information: The Course outline will be made available closer to the start of term - please visit this website: www.unsw.edu.au/course-outlines
The Course Outline contains information about course objectives, assessment, course materials and the syllabus.
Important additional information as of 2023
UNSW Plagiarism Policy
The University requires all students to be aware of its policy on plagiarism.
For courses convened by the School of Mathematics and Statistics no assistance using generative AI software is allowed unless specifically referred to in the individual assessment tasks.
If its use is detected in the no assistance case, it will be regarded as serious academic misconduct and subject to the standard penalties, which may include 00FL, suspension and exclusion.
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.
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.