Capacitability theorem in measurable gambling theory

Abstract:

A player in a measurable gambling house $\Gamma$ defined on a Polish state space $X$ has available, for each $x \in X$, the collection $\Sigma (x)$ of possible distributions $\sigma$ for the stochastic process ${x_1},{x_2}, \ldots$ of future states. If the object is to control the process so that it will lie in an analytic subset $A$ of $H = X \times X \times \cdots$, then the player’s optimal reward is \[ M(A)(x) = \sup \{ \sigma (A):\sigma \in \Sigma (x)\}.\] The operator $M( \bullet )(x)$ is shown to be regular in the sense that \[ M(A)(x) = \inf M(\{ \tau < \infty \} )(x),\] where the infimum is over Borel stopping times $\tau$ such that $A \subseteq \{ \tau < \infty \}$. A consequence of this regularity property is that the value of $M(A)(x)$ is unchanged if, as in the gambling theory of Dubins and Savage, the player is allowed to use nonmeasurable strategies. This last result is seen to hold for bounded, Borel measurable payoff functions including that of Dubins and Savage.