Game-Theoretic Foundations for Probability and Finance

9Relation to Measure‐Theoretic Probability

As we noted in the preface, the game‐theoretic and measure‐theoretic pictures are equivalent for coin tossing but generalize this core example in different directions. We have also noted in passing various other ways in which the two pictures overlap and complement each other. As we see in this chapter, we can connect them more systematically, recasting game‐theoretic martingales and supermartingales as measure‐theoretic martingales and supermartingales or vice versa. This can permit the ready derivation of many results in one picture from corresponding results in the other.

Measure‐theoretic probability has two distinct ways of representing a sequence of outcomes. In one, the probability space is canonical, as it consists of the different possibilities for how the sequence can come out. In the other, successive outcomes are treated as random variables. The canonical representation comes closest to game‐theoretic probability, at least as presented in this book in the case of discrete time. So we consider it first, in Sections 9.1 and 9.2. We consider the more general noncanonical representation in Section 9.3. In each case, we use a law of large numbers to illustrate how a game‐theoretic result can be translated into a measure‐theoretic result or vice versa.

Long before measure‐theoretic probability was put in its modern form, mathematicians knew how to construct probabilities for a finite or infinite sequence of outcomes by combining ...