Saturations of gambling houses

Abstract

Suppose that X is a Borel subset of a Polish space. Let ℙ(X) be the set of probability measures on the Borel σ-field of X. We equip ℙ(X) with the weak topology. A gambling house Γ on X is a subset of X × ℙ(X) such for each x ε X, the section Γ(x) of Γ at x is nonempty. Assume moreover that Γ is an analytic subset of X × ℙ(X). Then we can associate with Γ optimal reward operators G _Γ, R _Γ, and M _Γ as follows:

$$\begin{gathered}(G_\Gamma u)(x) = \sup \{ \smallint ud\gamma :\gamma \in \Gamma (x)\} ,x \in X, \hfill \\(R_\Gamma u)(x) = \sup \int {u(x_t )dP_\sigma ,x \in X} \hfill \\\end{gathered}$$

where u is a bounded, Borel measurable function on X, the sup in the definition of R _Γ is over all measurable strategies σ available in Γ at x and Borel measurable stop rules t (including t ≡ 0), x _t is the terminal state and P _σ the probability measure on H, the space of infinite histories, induced by σ;

$$(M_\Gamma g)(x) = \sup \int {gdP_\sigma ,x \in X,}$$

where g is a bounded, Borel measurable function on H and the sup is over all measurable strategies σ available in Γ at x. The aim of this article is to describe the “largest” houses or “saturations” for which the associated operators are the same as the corresponding operators for the original house. Our methods are constructive and will show that the saturations are again analytic gambling houses.

Research supported by National Science Foundation Grant DMS-9703285.