Probability Measures on Metric Spaces

This chapter leaves the finite-dimensional framework, because it deals with probability measures on the sigma-field of Borel sets in a general metric space. As a start, there is a motivation to go beyond the finite-dimensional case. To this end, you learn the uniform empirical process, and you get to know that the famous Kolmogorov-Smirnov- and the Cramér-von Mises statistics are functionals of this process. The chapter proceeds with a compilation of basic notions and results in connection with metric spaces, such as open and closed sets, convergence, separability, completeness, basis, relative compactness, and Baire’s category theorem. A main focus is on the space C[0,1] of real-valued continuous functions defined on the unit interval, equipped with the supremum distance. In this space, relative compact subsets are characterized by the Arzelà-Ascoli theorem. The sigma-field of Borel sets in C[0,1] is generated by the field of finite-dimensional sets. Further basic terms introduced in this chapter are separating class, coordinate projection, and modulus of continuity.