MTH5099 - Measure theory - 2019

6 points, SCA Band 2, 0.125 EFTSL

Refer to the specific census and withdrawal dates for the semester(s) in which this unit is offered.

Faculty

Science

Organisational Unit

School of Mathematical Sciences

Chief examiner(s)

Coordinator(s)

Unit guides

Offered

Clayton

• First semester 2019 (On-campus)

Prerequisites

Enrolment in the Master of Mathematics

MTH3140

Prohibitions

MTH4099

Synopsis

Measure theory is one of the few theories which permeates all core mathematical domains (pure, applied and statistics). We develop Lebesgue integration and probability theory from the core elements of measure theory. The initial background will be kept to a minimum. In particular, it is only required knowledge of real analysis and elementary probability theory (prior knowledge of functional analysis is not required, but it is definitely encouraged). On the other hand, the topics covered in this course will be fundamental for the understanding of advanced courses (differential geometry, advanced analysis, partial differential equations), as described above.

The unit will cover such pure topics as: semi-rings, algebras, and sigma-algebras of sets, measures, outer measures, the Lebesgue and Borel measures, construction of Vitali sets, construction of non-Borel Lebesgue measurable sets, measurable and integrable functions, the Lebesgue integral and the fundamental theorems, change of variables formula in Euclidean space, the Lebesgue spaces, iterated measures and the Fubini theorem, modes of convergence, signed measures, decomposition of measures and the Radon-Nikodym theorem, approximation results for the Lebesgue measure, Hausdorff measure and dimension, Haar measures, ergodic measures.

The unit will also cover topics which are essential for probability theory: such as Borel-Cantelli Lemma, independence, Kolmogorov 0-1 law, exponential bounds, conditional expectation, martingales.

Outcomes

On successful completion of this unit, students will be able to:

1. Formulate complex problems using appropriate measure theory terminology.
2. Use sophisticated tools from measure theory in various areas of Mathematics (e.g. partial differential equations, geometric analysis, dynamical systems, general relativity, probability theory).
3. Identify specific situations to which the fundamental results of measure theory apply, and demonstrate advanced expertise in applying these results to said situations.
4. Communicate complex results and specialised information using the language of measure theory.

Assessment

Examination (3 hours): 60% (Hurdle)

Continuous assessment: 40%

Hurdle requirement: To pass this unit a student must achieve at least 50% overall and at least 40% for the end-of-semester exam.

This unit is offered at both Level 4 and Level 5, differentiated by the level of the assessment. Students enrolled in MTH5099 will be expected to demonstrate a higher level of learning in this subject than those enrolled in MTH4099. The assignments and exam in this unit will use some common items from the MTH4099 assessment tasks, in combination with several higher level questions and tasks.