A Borel Measurable Version of Konig's Lemma for Random Paths

Abstract

Starting at $x$ in a Polish space $X$, a player selects the distribution $\sigma_0$ of the next state $x_1$ from the collection $\Gamma (x)$ of those distributions available and then selects the distribution $\sigma_1(x_1)$ for $x_2$ from $\Gamma(x_1)$ and so on. Suppose the player wins if every $x_i$ in the stochastic process $x_1, x_2,\ldots$ lies in a given Borel subset $A$ of $X$, that is, if the process stays in $A$ forever. If $\{(x, \gamma): \gamma \in \Gamma (x)\}$ is a Borel subset of $X \times \mathbb{P}(X)$, where $\mathbb{P}(X)$ is the natural Polish space of probability measures on $X$, and if $0 \leq p \leq 1$, then a player can stay in $A$ forever with probability at least $p$ if and only if the player can stay in $A$ up to time $t$ with probability at least $p$ for every Borel stop rule $t$. A similar result holds when the object of the game is to visit $A$ infinitely often.