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Unit of study_

MATH4069: Measure Theory and Fourier Analysis

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.

Code MATH4069
Academic unit Mathematics and Statistics Academic Operations
Credit points 6
(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 MATH3978 or MATH3979)]
Assumed knowledge:
(MATH2921 and MATH2922) or MATH2961

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.