Integration and Analysis III
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The Riemann integral works well for continuous functions on closed bounded intervals, but it has certain deficiencies that cause problems, for example, in Fourier analysis and in the theory of differential equations. To overcome such deficiencies, a "new and improved" version of the integral was developed around the beginning of the twentieth century, and it is this theory with which this course is concerned. The underlying basis of the theory, measure theory, has important applications not just in analysis but also in the modern theory of probability.
To introduce students to the modern approach to
integration as developed notably by Lebesgue and to enable them to
understand how this theory overcomes some of the main inadequacies of
"standard" Riemann integration and to appreciate why it is important
to be able to do this. The course also aims to provide a strong base
from which students wishing to purue studies in areas such as
analysis, physics, probability theory, financial mathematics and risk
analysis will be able to proceed. At the end of this course students should be
understand what Lebesgue measure is;
understand what measurable sets and functions are;
understand the construction of the Lebesgue integral and its
relation to the Riemann integral;
be able to integrate a variety of different functions using the
definition of the Lebesgue integral;
be familiar with the basic convergence theorems of measure
understand the Fubini and Tonelli theorems;
understand what the classical Lp spaces are, and know some of
their most important properties.
Topics covered are: Set theory; Lebesgue outer measure; measurable sets; measurable functions. Integration of measurable functions over measurable sets. Convergence of sequences of functions and their integrals. General measure spaces and product measures. Fubini and Tonelli's theorems. Lp spaces. The Radon-Nikodym theorem. The Riesz representation theorem. Integration and Differentiation.
This course provides a foundation for further
studies in mathematical analysis, mathematical physics, probability
theory, risk analysis, financial mathematics and mathematical