Probability Spaces

Abstract

The fundamentals of probability spaces are developed: sample spaces, event spaces or σ-fields, and probability measures. The emphasis is on sample spaces that are Polish spaces — complete, separable metric spaces — and on Borel σ-fields — event spaces generated by the open sets of the underlying metric space. Complete probability spaces are considered in order to facilitate later comparison between standard spaces and Lebesgue spaces. Outer measures are used to prove the Carathéodory extension theorem, providing conditions under which a candidate probability measure on a field extends to a unique probability measure on a σ-field.