# MATHS 730 : Measure Theory and Integration

## Science

### Course Prescription

Presenting the modern elegant theory of integration as developed by Riemann and Lebesgue, it includes powerful theorems for the interchange of integrals and limits so allowing very general functions to be integrated, and illustrates how the subject is both an essential tool for analysis and a critical foundation for the theory of probability. Strongly recommended: MATHS 333.

### Course Overview

This course develops the basic theory of measure and integration, with Lebesgue measure as a motivating example.
It includes powerful theorems for the interchange of integrals and limits, and allows very general functions to be integrated. It provides the essential tools for analysis and applied mathematics, and is the foundation for the theory of probability and parts of functional analysis.

### Course Requirements

Prerequisite: MATHS 332

### Capabilities Developed in this Course

 Capability 1: Disciplinary Knowledge and Practice Capability 2: Critical Thinking Capability 3: Solution Seeking Capability 4: Communication and Engagement Capability 5: Independence and Integrity

### Learning Outcomes

By the end of this course, students will be able to:
1. To know the basic definitions of measure and integral. This includes an intuitive understanding, from which the definitions could essentially be recalled many years after the course. (Capability 1 and 2)
2. To know the basic theory of measure and integral. This is to be learned to high level during the course, and after the basic gist of application and arguments should be maintained. (Capability 1, 2 and 3)
3. To be able to solve problems involving in measure theory. As this a completely rigorous course, it allows for the presentation of mathematical arguments in the assignments. It is generally expected that students will make substantive progress in proof presentation in the course. Usually this involves going from overly long "proofs", often with incorrect arguments, to simple arguments with just the right amount detail to give a convincing proof. (Capability 1, 2, 3, 4 and 5)
4. A high level of in class participation. It is expected that students, will ask questions and take an active part in lectures. Sometimes this involves students (either individually or as a pair) giving short presentations of some applications that cannot be covered in the theorem/proof detail of the rest of the course. (Capability 1, 2, 3, 4 and 5)

### Assessments

Assessment Type Percentage Classification
Assignments 30% Individual Coursework
Final Exam 70% Individual Examination
1 2 3 4
Assignments
Final Exam

plussage

### Learning Resources

The Elements of Integration and Lebesgue Measure, R. G. Bartle, is often given as a recommended text book.

### Special Requirements

No special requirements, though completion of assignments is highly recommended.

This course is a standard 15 point course and students are expected to spend 10 hours per week involved in each 15 point course that they are enrolled in.

For this course, you can expect 3 hours of lectures, 2 hours of reading and thinking about the content and 5 hours of work on assignments and/or test preparation.

### Digital Resources

Course materials are made available in a learning and collaboration tool called Canvas which also includes reading lists and lecture recordings (where available).

Please remember that the recording of any class on a personal device requires the permission of the instructor.

The content and delivery of content in this course are protected by copyright. Material belonging to others may have been used in this course and copied by and solely for the educational purposes of the University under license.

You may copy the course content for the purposes of private study or research, but you may not upload onto any third party site, make a further copy or sell, alter or further reproduce or distribute any part of the course content to another person.

The University of Auckland will not tolerate cheating, or assisting others to cheat, and views cheating in coursework as a serious academic offence. The work that a student submits for grading must be the student's own work, reflecting their learning. Where work from other sources is used, it must be properly acknowledged and referenced. This requirement also applies to sources on the internet. A student's assessed work may be reviewed against online source material using computerised detection mechanisms.

### Inclusive Learning

All students are asked to discuss any impairment related requirements privately, face to face and/or in written form with the course coordinator, lecturer or tutor.

Student Disability Services also provides support for students with a wide range of impairments, both visible and invisible, to succeed and excel at the University. For more information and contact details, please visit the Student Disability Services’ website at http://disability.auckland.ac.nz

### Special Circumstances

If your ability to complete assessed coursework is affected by illness or other personal circumstances outside of your control, contact a member of teaching staff as soon as possible before the assessment is due.

If your personal circumstances significantly affect your performance, or preparation, for an exam or eligible written test, refer to the University’s aegrotat or compassionate consideration page: https://www.auckland.ac.nz/en/students/academic-information/exams-and-final-results/during-exams/aegrotat-and-compassionate-consideration.html.

This should be done as soon as possible and no later than seven days after the affected test or exam date.

### Student Feedback

During the course Class Representatives in each class can take feedback to the staff responsible for the course and staff-student consultative committees.

At the end of the course students will be invited to give feedback on the course and teaching through a tool called SET or Qualtrics. The lecturers and course co-ordinators will consider all feedback.

Your feedback helps to improve the course and its delivery for all students.

### Student Charter and Responsibilities

The Student Charter assumes and acknowledges that students are active participants in the learning process and that they have responsibilities to the institution and the international community of scholars. The University expects that students will act at all times in a way that demonstrates respect for the rights of other students and staff so that the learning environment is both safe and productive. For further information visit Student Charter (https://www.auckland.ac.nz/en/students/forms-policies-and-guidelines/student-policies-and-guidelines/student-charter.html).

### Disclaimer

Elements of this outline may be subject to change. The latest information about the course will be available for enrolled students in Canvas.

In this course you may be asked to submit your coursework assessments digitally. The University reserves the right to conduct scheduled tests and examinations for this course online or through the use of computers or other electronic devices. Where tests or examinations are conducted online remote invigilation arrangements may be used. The final decision on the completion mode for a test or examination, and remote invigilation arrangements where applicable, will be advised to students at least 10 days prior to the scheduled date of the assessment, or in the case of an examination when the examination timetable is published.

Published on 14/02/2020 09:08 p.m.