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Unit of study_

MATH3969: Measure Theory and Fourier Analysis (Adv)

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.

Code MATH3969
Academic unit Mathematics and Statistics Academic Operations
Credit points 6
A mark of 65 or greater in 12 credit points of MATH2XXX units of study
Assumed knowledge:
Real analysis and vector spaces. For example MATH2X21 and MATH2X23

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.