# MATHS 730 : Measure Theory and Integration

## Science

### Course Prescription

Presents the modern elegant theory of integration as developed by Riemann and Lebesgue. This course includes powerful theorems for the interchange of integrals and limits, 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 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 3: Knowledge and Practice Capability 4: Critical Thinking Capability 5: Solution Seeking Capability 6: Communication

### Learning Outcomes

By the end of this course, students will be able to:
1. 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 3, 4 and 6)
2. Solve problems involving 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 3, 4 and 5)
3. Display / continue to build on their mathematical maturity. It is expected that students, will ask questions and take an active part in lectures. 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. In assessments students should display their mathematical maturity by being able to approach unfamiliar problems and critically evaluate purported proofs/examples. (Capability 3, 4 and 5)
4. 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 3, 4, 5 and 6)

### Assessments

Assessment Type Percentage Classification
Assignments 30% Individual Coursework
Final Exam 70% Individual Coursework
1 2 3 4
Assignments
Final Exam
The final grade for each student will be calculated using the plussage system, taking the maximum of two scores: EITHER 30% from assignments and 70% from the final exam, OR 100% from the final exam.

### Tuākana

Tuākana Science is a multi-faceted programme for Māori and Pacific students providing topic specific tutorials, one-on-one sessions, test and exam preparation and more. Explore your options at
https://www.auckland.ac.nz/en/science/study-with-us/pacific-in-our-faculty.html
https://www.auckland.ac.nz/en/science/study-with-us/maori-in-our-faculty.html

Whanaungatanga and manaakitanga are fundamental principles of our Tuākana Mathematics programme which provides support for Māori and Pasifika students who are taking mathematics courses. The Tuākana Maths programme consists of workshops and drop-in times, and provides a space where Māori and Pasifika students are able to work alongside our Tuākana tutors and other Māori and Pasifika students who are studying mathematics.

### Key Topics

The first half of the course will cover the general theory:

• Introduction, set theory: cardinality and de Morgan's laws
• The extended reals, and infinite limits of sequences
• The liminf (limit inferior) and limsup (limit superior) of a sequence of extended reals, and their properties.
• sigma algebras, and the example of the Borel sigma algebra
• measurable functions
• sums, products, limits, limsups, etc of measurable functions
• measures, and a proof there is no measure on the power set of the reals which has the basic properties we would desire
• the measure of expanding and contracting sequences of sets, and sets of measure zero, including the Cantor set
• Simple function, the approximation of a nonnegative measurable function by an increasing sequence of simple functions
• The integral of a simple nonnegative function
• Integration of nonnegative measurable functions, and the monotone convergence theorem
• Fatou's Lemma, integration against a nonnegative measurable function gives a measure.
• The integral of a nonnegative function is zero iff it is zero a.e. Definition of the integral of f in terms of the integral of its positive and negative parts and space L of integrable functions.
• Dominated convergence theorem, and the continuous version.
• Norms, and L_p spaces (everything but Minkowskii's inequality and completeness proved).
• L_p is a Banach space.
in full detail. More specialised topics, including further examples of measures, will be covered in the second half.

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 each week of 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.

### Delivery Mode

#### Campus Experience

• Attendance at lectures is expected.
• The activities for the course are scheduled as a standard weekly timetable.

### Learning 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.

Recommended text:

• The Elements of Integration and Lebesgue Measure, R. G. Bartle.

### 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.

The application of fractals may be added (to the general course), based on the lectures done by Melissa Tacy (a specialist in this area).

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 for potential plagiarism or other forms of academic misconduct, using computerised detection mechanisms.

### Class Representatives

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.

### 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

### 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 .

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

### Learning Continuity

In the event of an unexpected disruption, we undertake to maintain the continuity and standard of teaching and learning in all your courses throughout the year. If there are unexpected disruptions the University has contingency plans to ensure that access to your course continues and course assessment continues to meet the principles of the University’s assessment policy. Some adjustments may need to be made in emergencies. You will be kept fully informed by your course co-ordinator/director, and if disruption occurs you should refer to the university website for information about how to proceed.

The delivery mode may change depending on COVID restrictions. Any changes will be communicated through Canvas.

### 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 .

### 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 students may be asked to submit 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. In exceptional circumstances changes to elements of this course may be necessary at short notice. Students enrolled in this course will be informed of any such changes and the reasons for them, as soon as possible, through Canvas.

Published on 31/10/2023 10:53 a.m.