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Overview

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.


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 if 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.

Workload Expectations

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.

Course Prerequisites, Corequisites and Restrictions

Prerequisite

Locations and Semesters Offered

LocationSemester
City

Teaching and Learning

Campus Experience

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

Learning Resources

Taught courses use a learning and collaboration tool called Canvas to provide students with learning materials including 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.

Additional Information on Learning Resources

Recommended text: 

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

Copyright

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.

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.

Academic Integrity

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.

Similarly, research students must meet the University’s expectations of good research practice. This requires:

  • Honesty - in all aspects of research work
  • Accountability - in the conduct of research
  • Professional courtesy and fairness – in working with others
  • Good stewardship – on behalf of others
  • Transparency – of research process and presentation of results
  • Clarity - communication to be understandable, explainable and accessible

For more information on the University’s expectations of academic integrity, please see the Academic Conduct section of the University policy hub.

Disclaimer

Elements of this outline may be subject to change. The latest information about taught courses is made available to enrolled students in Canvas.

Students may be asked to submit assessments digitally. The University reserves the right to conduct scheduled tests and examinations 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.


Assessment and Learning Outcomes

Additional Information on Assessment

The final grade for each student will be calculated using the plussage system, taking the maximum of two scores: EITHER 50% from assignments and 50% from the final exam, OR 100% from the final exam.

Special Requirements

Students must collect their marked assignments.

Course Learning Outcomes

CLO #OutcomeProgramme Capability Link
1
2
3
4

Assessments

Assessment TypeAssessment PercentageAssessment Classification

Assessment to CLO Mapping

Assessment Type1234

Student Feedback, Support and Charter

Student Feedback

Feedback on taught courses is gathered from students at the end of each semester through a tool called SET or Qualtrics. The lecturers and course co-ordinators will consider all feedback and respond with summaries and actions. Your feedback helps teachers to improve the course and its delivery for future students. In addition, class Representatives in each class can take feedback to the department and faculty staff-student consultative committees.

Additional Information on Student Feedback

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

Class representatives

Class representatives are students tasked with representing student issues to departments, faculties, and the wider university. If you have a complaint about this course, please contact your class rep who will know how to raise it in the right channels. See your departmental noticeboard for contact details for your class reps.

Tuākana

Tuākana 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 Tuakana Learning Communities.

Additional Information on Tuākana

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.

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.

Wellbeing

We all go through tough times during the semester, or see our friends struggling. There is lots of help out there - please see the Support Services page for information on support services in the University and the wider community.

Special Circumstances

If your ability to complete assessed work 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.

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.

Student Academic Complaints and Disputes

Students with concerns about teaching including how a course is delivered, the resources provided, or supervision arrangements, have the right to express their concerns and seek resolution. The university encourages informal resolution where possible, as this is quicker and less stressful. For information on the informal and formal complaints processes, please refer to the Student Academic Complaints Statute in the Student Policies and Guidelines section of the Policy Hub.