Metric Spaces

In this module,students will construct Lebesgue measure on the line, extending the idea of the length of an interval. They will use this to define an integral which is shown to have good properties under pointwise convergence. Looking at some basic results about the set of real numbers, students will explore properties of countable sets, open sets and algebraic numbers.T

They will also have the opportunity to illustrate the power of the convergence theorems in applications to some classical limit problems and analysis of Fourier integrals, which are fundamental to probability theory and differential equations.