Preliminary Material: Extension Theorems, Martingales, and Compactness

Abstract

As mentioned in the Introduction, the point of view that we take will involve us in a detailed study of measures on function spaces. There are a few basic tools which are necessary for the construction of such measures. The purpose of this chapter is to develop these tools. In the process, we will introduce some notions (e.g., conditioning and martingales) which will play an important role in what follows. Section 1.1 contains the basic theorem of Prohorov and Varadarajan characterizing weakly compact families of measures on a Polish space. Using their results, we prove the existence of conditional probability distributions. The final topics in Section 1.1 are the extension theorems of Tulcea and Kolmogorov. Section 1.2 introduces the notions of progressively measurable functions and martingales. In connection with martingales we prove Doob’s inequality, his stopping time theorem and a useful integration by parts formula. Finally we prove a result connecting martingale theory and conditioning.