MATH31011/41011 - 2010/2011

 

Specification

Aims

To give an introduction to Lebesgue's theory of measure and integration on the set of real numbers R. To use this to find an appropriate setting in which to understand the convergence of Fourier series.

Brief Description of the unit

It is often convenient to represent functions as Fourier series. However, the convergence of such series is a delicate issue closely related to the theory of integration. A standard approach to integration on the real line, formalised by Riemann, is based on partitioning the domain into smaller intervals. (This theory was described in MATH20101 but is not a prerequisite for the course.) This approach works in many situations but there are simple examples for which it fails. In the early 1900s, H. Lebesgue produced a better  theory in which the key idea is to extend the notion of length from intervals to more complicated subsets of R. This started an area of mathematics it its own right, called Measure Theory. Most generally, this is about how one may sensibly assign a size to members of a collection of sets. One application of Lebesgue's ideas is that one can introduce a vector space of functions in which Fourier series appear in a natural way.

This course will appeal to students who have enjoyed MATH20101 or MATH20111 and MATH20122. It will be useful to student taking probability course courses in years three and four since the ideas of measure theory have a central role in probability theory.

Learning Outcomes

On successful completion of this course unit students will

• understand how Lebesgue measure on R is defined,
• understand how measures may be used to construct integrals,
• know the basic convergence theorems for the Lebesgue integral,
• understand the relation between Fourier series and the Hilbert space of square integrable functions.

Future topics requiring this course unit

It would be helpful for level 3 and 4 courses in probability.

Syllabus

1. Fourier series, convergence and Dirichlet's Theorem, the Gibbs phenomenon. [4 lectures]
2. Revision of countable and uncountable sets, the Cantor set. [2 lectures]
3. Riemann's approach to integration. Lesbesgue measure on R, Borel sets, measurable sets and functions, construction of the Lebesgue integral. [8 lectures]
4. Limit theorems for the Lebesgue integral. [2 lectures]
5. Square integrable functions and Fourier series, Hilbert spaces. [6 lectures]

For MATH41011 the lectures will be enhanced by additional reading on the costruction of Lebesgue measure, existence of non-measurable sets, and general measures. Reading material will be provided.

Textbooks

• J. Franks, A (Terse) Introduction to Lebesgue Integration, American Mathematical Society, Student Mathematical Library, 2009.
• H. S. Bear, A Primer of Lebesgue Integration, Academic Press, 1995.

Teaching and learning methods

Two lectures and one examples class each week. In addition students should expect to spend at least four hours each week on private study for this course unit (seven hours for MATH41011).

Assessment

End of semester examination: two hours weighting 100% (MATH31011), three hours weighting 100% (MATH41011)

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