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

Timetable

Lecturer

Daniel Daners, Carslaw Room 715, phone 9351 2966
email: daniel.daners@sydney.edu.au
Consultation: Wednesday 13:00-14:00 or by appointment.

Lecture Notes

Daniel Daners, "Measure Theory and Fourier Analysis" (available from resources page). 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 handouts 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 one assignment worth 10%, and two 40 min quizzes counting 15% each. The final exam counts 60% of the total assessment.

Quiz Dates:

Quiz 1, 29 Aug in Tutorial
Quiz 2, 24 October in Tutorial

Assignment Due Date:

Friday 12 October

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

Late assignments are not accepted without prior arrangement well before the deadline!

Final Exam

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 and conditional probability., applications of the Radon-Nikodym theorem to conditional expectation.

Outcomes

• know the basics of abstract measure and integration theory
• has a good idea on the construction of measures from outer measures with its application to the Lebesgue measure
• 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 convolution.
• 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 the measure theoretic foundations of probability theory
• knows the definition and basic properties of conditional expectation

Reference Books

• HL Royden, Real Analysis, Macmillan, 1968.
• W Rudin, Real and complex analysis, McGraw-Hill, 1974

Books with view on probability:

• H Bauer, Probability theory and elements of measure theory, Academic Press, 1981.
• P Billingsley, Probability and Measure, Wiley, 1995.

More advanced books:

• L Evans and RF Gariepy, Measure theory and fine properties of functions, CRC Press, 1992.