# Unit Information for MATH3969: Measure Theory and Fourier Analysis (Advanced)

## Lecturer

Lecturer: Daniel Daners , email: MATH3969@maths.usyd.edu.au
Consultation: Thursday 13:00-14:00 or by appointment.

## Lecture Notes

Daniel Daners, "Measure Theory and Fourier Analysis" (available from Kopystop). You are also encouraged to make use of reference books (you do not need to buy any of these).

## Tutorials

Tutorial sheets are available online as PDF files from the resources page on the Friday of the week before the tutorial takes place. No hard copies will be distributed.

Tutorials are an integral part of the course. You can only learn mathematics (or any other subject) by doing problems yourself, so attending tutorials is absolutely essential for performing well in the course.

## Assessment

There will be two assignment worth 10%, and two 45 min quizzes counting 10% each. The final exam counts 60% of the total assessment.

Quiz Dates:
Quiz 1, Tuesday 22 August 9-10am in New Law 440
Quiz 2, Tuesday 17 October 9-10am in New Law 440
Assignment Due Date: (by midnight through LMS)
Assignment 1, Monday 11 September
Assignment 2, Monday 30 October

Note: The quizzes are held during the Tuesday tutorial for all students. If you cannot possibly make that time free contact the lecturer. Everyone is invited to join the Wednesday tutorial in the quiz week for a proper tutorial.

Group assignments are not permitted. You are encouraged to collaborate with others in solving the problems, but the work submitted must be your own!

The assignments must be submitted through the LMS, where they will be passed through the text matching software Turnitin (scanned copies of handwritten assignments are fine).

Late assignments are not accepted and no credit will be awarded

Final Exam: There will be a two-hour final exam. Only material covered in lectures and tutorials will be tested using questions addressing the outcomes. The exam will also contain questions on the theory and proofs, and not just problems to solve.

For all assessments, the rules for special consideration/arrangement apply. The maximal possible extension is 7 days.

The final mark is determined by the following criteria

• High Distinction (HD), 85–100: representing complete or close to complete mastery of the material;
• Distinction (D), 75–84: representing excellence, but substantially less than complete mastery;
• Credit (CR), 65–74: representing a creditable performance that goes beyond routine knowledge and understanding, but less than excellence;
• Pass (P), 50–64: representing at least routine knowledge and understanding over a spectrum of topics and important ideas and concepts in the course.
• Fail (F), 0–49: representing rather limited understanding on a significant range of topics and concepts.
There will be a two-hour final exam. Only material covered in lectures and tutorials will be tested.

## The Course

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. Probability Theory is then discussed. Definition and main properties of conditional expectation is given

Assumed Knowledge:Know the basics of real analysis and metric spaces (MATH2962 or MATH3961 are sufficient).

Outline
• Motivation for measure theory. Problems with an overly naive approach. Definition of measures and sigma algebras.
• Construction of measure on the real line and $$\mathbb R^N$$.
• Measurable functions, approximation by simple functions.
• Definition and first properties of integration, via integration of nonnegative functions and the Monotone Convergence Theorem.
• Integrating real and complex valued measurable funcions. The Dominated Convergence Theorem. Application to concrete examples such the product formula for the Gamma function.
• Product measures and double integrals.
• The $$L^p$$-spaces. Hölder's inequality, Minkowski's inequality, completeness.
• Convolution, approximate identities; Application to density theorems.
• The Fourier transform. Basic properties and examples. A simple inversion formula proved as an application of the Riemann-Lebesgue Lemma.
• Fourier tranforms and convolution. Application to inversion formulas for the Fourier transform.
• The Fourier transform on $$L^2$$. The Plancherel formula.
• Comparison of measures. Absolute continuity and the Radon-Nikodym theorem.
• The measure-theoretical basis for probability, distribution and distribution functions and densities.
• Applications of the Radon-Nikodym theorem to conditional expectation, basic properties of conditional expectation.

## Outcomes

• be familiar with the basics of abstract measure and integration theory
• has a good idea on the construction of measures from outer measures with application to the Lebesgue measure and related measures
• knows and be able to apply the limit theorems, in particular the monotone convergence theorem, the dominated convergence theorem and the theorems on continuity and differentiability of parameter integrals
• Knows the basic properties of $$L^p$$-spaces, their completeness and density theorems as well as properties of convolution.
• is able to work with inequalities such as Hölder's, Minkowski's, Jensen's and Young's inequality
• knows the basic properties of the Fourier transform on $$L^1$$ and $$L^2$$, including including the Riemann-Lebesgue lemma and Plancherel's theorem
• understands and is able to apply the measure theoretic foundations of probability theory, including distributions, distribution functions and densities
• knows the definition and basic properties of conditional expectation
• Is able to find and write simple proofs, and apply the theory in a number of applications

## Reference Books

Books with view on probability: