## The Annals of Probability

- Ann. Probab.
- Volume 19, Number 1 (1991), 423-451.

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

A. Maitra, R. Purves, and W. Sudderth

#### 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.

#### Article information

**Source**

Ann. Probab., Volume 19, Number 1 (1991), 423-451.

**Dates**

First available in Project Euclid: 19 April 2007

**Permanent link to this document**

https://projecteuclid.org/euclid.aop/1176990554

**Digital Object Identifier**

doi:10.1214/aop/1176990554

**Mathematical Reviews number (MathSciNet)**

MR1085346

**Zentralblatt MATH identifier**

0722.60037

**JSTOR**

links.jstor.org

**Subjects**

Primary: 60G40: Stopping times; optimal stopping problems; gambling theory [See also 62L15, 91A60]

Secondary: 93E20: Optimal stochastic control 04A15

**Keywords**

Measurable gambling optimization stop rules Konig's lemma analytic sets

#### Citation

Maitra, A.; Purves, R.; Sudderth, W. A Borel Measurable Version of Konig's Lemma for Random Paths. Ann. Probab. 19 (1991), no. 1, 423--451. doi:10.1214/aop/1176990554. https://projecteuclid.org/euclid.aop/1176990554