This chapter extends the concepts of convergence in distribution, in probability, and almost surely from Euclidean spaces to more abstract metric spaces. We are particularly interested in developing the theory for random functions, or stochastic processes, viewed as elements of the metric space of all bounded functions.
In this section we recall some basic topological concepts and introduce a number of examples of metric spaces.
A metric space is a set equipped with a metric. A metric or distance function is a map with the properties
(i) d(x, y) = ...