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Unit of study_

# MATH3969: Measure Theory and Fourier Analysis (Adv)

## Overview

Measure theory is the study of such fundamental ideas as length, area, volume, arc length and surface area. It is the basis for the integration theory used in advanced mathematics since it was developed by Henri Lebesgue in about 1900. Moreover, it is the basis for modern probability theory. The course starts by setting up measure theory and integration, establishing important results such as Fubini's Theorem and the Dominated Convergence Theorem which allow us to manipulate integrals. This is then applied to Fourier Analysis, and results such as the Inversion Formula and Plancherel's Theorem are derived. The Radon-Nikodyn Theorem provides a representation of measures in terms of a density. Probability theory is then discussed with topics including distributions and conditional expectation.

### Unit details and rules

Unit code MATH3969 Mathematics and Statistics Academic Operations 6 MATH4069 A mark of 65 or greater in 12 credit points of MATH2XXX units of study None Real analysis and vector spaces. For example MATH2X21 and MATH2X23 Yes

### Teaching staff

Coordinator Daniel Daners, daniel.daners@sydney.edu.au Daniel Daners

## Assessment

Type Description Weight Due Length
Final exam (Open book) Final exam
Written calculations
60% Formal exam period 2 hours
Outcomes assessed:
Assignment Quiz 1
Written calculations and arguments
10% Week 05
Due date: 24 Sep 2020 at 23:59

Closing date: 24 Sep 2020
45 minutes
Outcomes assessed:
Assignment Assignment 1
Written calculations and arguments
10% Week 07
Due date: 15 Oct 2020 at 23:59

Closing date: 25 Oct 2020
14 days
Outcomes assessed:
Assignment Quiz 2
Written calculations and arguments
10% Week 10
Due date: 05 Nov 2020 at 23:59

Closing date: 05 Nov 2020
45 minutes
Outcomes assessed:
Assignment Assignment 2
Written calculations and arguments
10% Week 12
Due date: 19 Nov 2020 at 23:59

Closing date: 29 Nov 2020
14 days
Outcomes assessed:
= Type C final exam

### Assessment summary

• 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.
• Quiz: 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.

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.

For more information see sydney.edu.au/students/guide-to-grades.

For more information see guide to grades.

### 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.

### Academic integrity

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.

## Learning support

### Simple extensions

If you encounter a problem submitting your work on time, you may be able to apply for an extension of five calendar days through a simple extension.  The application process will be different depending on the type of assessment and extensions cannot be granted for some assessment types like exams.

### Special consideration

If exceptional circumstances mean you can’t complete an assessment, you need consideration for a longer period of time, or if you have essential commitments which impact your performance in an assessment, you may be eligible for special consideration or special arrangements.

Special consideration applications will not be affected by a simple extension application.

### Using AI responsibly

Co-created with students, AI in Education includes lots of helpful examples of how students use generative AI tools to support their learning. It explains how generative AI works, the different tools available and how to use them responsibly and productively.

## 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. determine the measure theoretic foundations of probability theory from known applications of measure theory
• LO8. recall and explain the definition and basic properties of conditional expectation
• LO9. create proofs and apply measure theory in familiar applications.

### Graduate qualities

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

Learning outcomes Graduate qualities
GQ1 GQ2 GQ3 GQ4 GQ5 GQ6 GQ7 GQ8 GQ9

## Responding to student feedback

This section outlines changes made to this unit following staff and student reviews.

No change

## Additional information

### 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
• Follow safety instructions in your manual and posted in laboratories
• In case of fire, follow instructions posted outside the laboratory door
• First aid kits, eye wash and fire extinguishers are located in or immediately outside each laboratory
• As a precautionary measure, it is recommended that you have a current tetanus immunisation. This can be obtained from University Health Service: unihealth.usyd.edu.au/

### Disclaimer

The University reserves the right to amend units of study or no longer offer certain units, including where there are low enrolment numbers.

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