Unit of study_

# MATH4069: Measure Theory and Fourier Analysis

## Overview

Measure theory is the study of fundamental ideas as length, area, volume, arc length and surface area. It is the basis for Lebesgue integration theory used in advanced mathematics ever since its development in about 1900. Measure theory is also a key foundation for modern probability theory. The course starts by establishing the basics of measure theory and the theory of Lebesgue integration, including important results such as Fubini's Theorem and the Dominated Convergence Theorem which allow us to manipulate integrals. These ideas are applied to Fourier Analysis which leads to results such as the Inversion Formula and Plancherel's Theorem. The Radon-Nikodyn Theorem provides a representation of measures in terms of a density. Key ideas of this theory are applied in detail to probability theory to provide a rigorous framework for probability which takes in and generalizes familiar ideas such as distributions and conditional expectation. When you complete this unit you will have acquired a new generalized way of thinking about key mathematical concepts such as length, area, integration and probability. This will give you a powerful set of intellectual tools and equip you for further study in mathematics and probability.

### Details

Academic unit Mathematics and Statistics Academic Operations MATH4069 Measure Theory and Fourier Analysis Semester 2, 2023 Normal day Camperdown/Darlington, Sydney 6

### Enrolment rules

 Prohibitions ? MATH3969 (A mark of 65 or greater in 12cp of MATH2XXX units of study) or [12cp from the following units (MATH3061 or MATH3066 or MATH3063 or MATH3076 or MATH3078 or MATH3961 or MATH3962 or MATH3963 or MATH3969 or MATH3971 or MATH3974 or MATH3976 or MATH3977 or MAT None (MATH2921 and MATH2922) or MATH2961 Yes

### Teaching staff and contact details

Coordinator Nathan Brownlowe, nathan.brownlowe@sydney.edu.au Nathan Brownlowe math3969@sydney.edu.au All email regarding the unit should be sent to this address. It goes to the lecturer and administrative staff.

## Assessment

Type Description Weight Due Length
Supervised exam

Final exam
Written calculations and arguments
60% Formal exam period 2 hours
Outcomes assessed:
Small test Quiz 1
Written calculations and arguments
10% Week -05
Due date: 30 Aug 2023 at 23:59

Closing date: 30 Aug 2023
45 minutes
Outcomes assessed:
Assignment Assignment 1
Written calculations and arguments
10% Week 07
Due date: 15 Sep 2023 at 23:59

Closing date: 25 Sep 2023
14 days
Outcomes assessed:
Small test Quiz 2
Written calculations
10% Week 10
Due date: 11 Oct 2023 at 23:59

Closing date: 11 Oct 2023
45 minutes
Outcomes assessed:
Assignment Assignment 2
Written calculations and arguments
10% Week 12
Due date: 27 Oct 2023 at 23:59

Closing date: 06 Nov 2023
14 days
Outcomes assessed:
• Assignments: There are two assignments. Each must be submitted electronically, as one single typeset or scanned PDF file only, via Canvas by the deadline. Note that your assignment will not be marked if it is illegible or if it is submitted sideways or upside down. It is your responsibility to check that your assignment has been submitted correctly and that it is complete (check that you can view each page). Late submisions will receive a penalty.
• Quizzes: Two quizzes will be held online through Canvas. Each quiz is 45 Minutes and has to be submitted by the closing time of 23:59 on the due date. The quiz can be taken any time during the 24 hour period before the closing time. The better mark principle will be used for the quiz so do not submit an application for Special Consideration or Special Arrangements if you miss a quiz. The better mark principle means that the quiz counts if and only if it is better than or equal to your exam mark. If your quiz mark is less than your exam mark, the exam mark will be used for that portion of your assessment instead.
• Final exam: The exam will cover material covered in lectures and tutorials including the theory and proofs, and not just problems to solve.

Final exam: If a second replacement exam is required, this exam may be delivered via an alternative assessment method, such as a viva voce (oral exam). The alternative assessment will meet the same learning outcomes as the original exam. The format of the alternative assessment will be determined by the unit coordinator

Detailed information for each assessment can be found on Canvas.

### Assessment criteria

The University awards common result grades, set out in the Coursework Policy 2014 (Schedule 1).

As a general guide, a high distinction indicates work of an exceptional standard, a distinction a very high standard, a credit a good standard, and a pass an acceptable standard.

Result name

Mark range

Description

High distinction

85 - 100

Representing complete or close to complete mastery of the material.

Distinction

75 - 84

Representing excellence, but substantially less than complete mastery.

Credit

65 - 74

Representing a creditable performance that goes beyond routine knowledge and understanding, but less than excellence.

Pass

50 - 64

Representing at least routine knowledge and understanding over a spectrum of topics and important ideas and concepts in the course.

Fail

0 - 49

When you don’t meet the learning outcomes of the unit to a satisfactory standard.

### Late submission

In accordance with University policy, these penalties apply when written work is submitted after 11:59pm on the due date:

• Deduction of 5% of the maximum mark for each calendar day after the due date.
• After ten calendar days late, a mark of zero will be awarded.

### Special consideration

If you experience short-term circumstances beyond your control, such as illness, injury or misadventure or if you have essential commitments which impact your preparation or performance in an assessment, you may be eligible for special consideration or special arrangements.

The Current Student website provides information on academic integrity and the resources available to all students. The University expects students and staff to act ethically and honestly and will treat all allegations of academic integrity breaches seriously.

We use similarity detection software to detect potential instances of plagiarism or other forms of academic integrity breach. If such matches indicate evidence of plagiarism or other forms of academic integrity breaches, your teacher is required to report your work for further investigation.

You may only use artificial intelligence and writing assistance tools in assessment tasks if you are permitted to by your unit coordinator, and if you do use them, you must also acknowledge this in your work, either in a footnote or an acknowledgement section.

Studiosity is permitted for postgraduate units unless otherwise indicated by the unit coordinator. The use of this service must be acknowledged in your submission.

## Weekly schedule

WK Topic Learning activity Learning outcomes
Week 01 Introduction, σ-algebras and measures and basic properties, outer measures and the Lebesgue measure Lecture (3 hr)
Week 02 More on the Lebesgue measure, measurable functions. Lecture and tutorial (4 hr)
Week 03 approximation of measurable functions by simple functions, definition of the abstract Lebesgue intergral, monotone convergence Lecture and tutorial (4 hr)
Week 04 Limit theorems including the dominated convergence theorem. Lecture and tutorial (4 hr)
Week 05 Integrals with parameters, introduction to L_p-spaces and related inequalities (Young, Hölder, Minkowski). Lecture and tutorial (4 hr)
Week 06 Convergence and completeness properties of L_p-spaces, essential supremum Lecture and tutorial (4 hr)
Week 07 Tonelli's and Fubini's theorem, continuity of translations on L^p and convolutions. Lecture and tutorial (4 hr)
Week 08 Convolution and approximate identities Lecture and tutorial (4 hr)
Week 09 Test functions and density in L_p, introduction of the Fourier transform on L_1. Lecture and tutorial (4 hr)
Week 10 Fourier inversion formula on L_1, Plancherel's theorem and the definition and propeties of the Fourier transform on L_2 Lecture and tutorial (4 hr)
Week 11 The Riesz representation theory in a Hilbert space, absolute continuity and the Radon-Nikodym theorem for the existence of densities. Lecture and tutorial (4 hr)
Week 12 Applications to probability theory: random variables, distributions, distribution functions and densities. An application of the Radon-Nikodym theorem to the construction of conditional expectation. Lecture and tutorial (4 hr)

### Study commitment

Typically, there is a minimum expectation of 1.5-2 hours of student effort per week per credit point for units of study offered over a full semester. For a 6 credit point unit, this equates to roughly 120-150 hours of student effort in total.

## Learning outcomes

Learning outcomes are what students know, understand and are able to do on completion of a unit of study. They are aligned with the University’s graduate qualities and are assessed as part of the curriculum.

At the completion of this unit, you should be able to:

• LO1. explain and apply the fundamentals of abstract measure and integration theory
• LO2. explain what an outer measure and use outer measures to construct other measures. Apply these concepts to Lesbesgue measures and other related measures.
• LO3. explain and apply the limit theorems including the dominated convergence theorem and theorems on continuity and differentiability of parameter integrals.
• LO4. explain the properties of Lp spaces
• LO5. use inequalities such as Holder’s, Minkowsi’s, Jensen’s and Young’s inequalities to solve problems
• LO6. explain and apply the properties of the Fourier series on normed spaces
• LO7. generate the measure theoretic foundations of probability theory
• LO8. recall and explain the definition and basic properties of conditional expectation
• LO9. create proofs and apply measure theory in diverse applications.

The graduate qualities are the qualities and skills that all University of Sydney graduates must demonstrate on successful completion of an award course. As a future Sydney graduate, the set of qualities have been designed to equip you for the contemporary world.

 GQ1 Depth of disciplinary expertise Deep disciplinary expertise is the ability to integrate and rigorously apply knowledge, understanding and skills of a recognised discipline defined by scholarly activity, as well as familiarity with evolving practice of the discipline. GQ2 Critical thinking and problem solving Critical thinking and problem solving are the questioning of ideas, evidence and assumptions in order to propose and evaluate hypotheses or alternative arguments before formulating a conclusion or a solution to an identified problem. GQ3 Oral and written communication Effective communication, in both oral and written form, is the clear exchange of meaning in a manner that is appropriate to audience and context. GQ4 Information and digital literacy Information and digital literacy is the ability to locate, interpret, evaluate, manage, adapt, integrate, create and convey information using appropriate resources, tools and strategies. GQ5 Inventiveness Generating novel ideas and solutions. GQ6 Cultural competence Cultural Competence is the ability to actively, ethically, respectfully, and successfully engage across and between cultures. In the Australian context, this includes and celebrates Aboriginal and Torres Strait Islander cultures, knowledge systems, and a mature understanding of contemporary issues. GQ7 Interdisciplinary effectiveness Interdisciplinary effectiveness is the integration and synthesis of multiple viewpoints and practices, working effectively across disciplinary boundaries. GQ8 Integrated professional, ethical, and personal identity An integrated professional, ethical and personal identity is understanding the interaction between one’s personal and professional selves in an ethical context. GQ9 Influence Engaging others in a process, idea or vision.

### Outcome map

GQ1 GQ2 GQ3 GQ4 GQ5 GQ6 GQ7 GQ8 GQ9

## Closing the loop

No changes have been made since this unit was last offered

### Work, health and safety

We are governed by the Work Health and Safety Act 2011, Work Health and Safety Regulation 2011 and Codes of Practice. Penalties for non-compliance have increased. Everyone has a responsibility for health and safety at work. The University’s Work Health and Safety policy explains the responsibilities and expectations of workers and others, and the procedures for managing WHS risks associated with University activities.

General laboratory safety rules

• No eating or drinking is allowed in any laboratory under any circumstances
• A laboratory coat and closed-toe shoes are mandatory