Let $X$ be a Polish space, i.e. a separable complete metric space. Any Borel probability measure on $X$ must be locally finite, outer regular and tight. Let $\mathcal{P}(X)$ be the set of all Borel probability measures on $X$.

A **rational preference relation** on $\mathcal{P}(X)$ is a binary relation $\precsim$ on $\mathcal{P}(X)$ that satisfies the following axioms:

**Completeness**: for any $ \mu, \nu \in \mathcal{P}(X) $, either $ \mu \precsim \nu $ or $ \nu \precsim \mu $**Transitivity**: for any $ \lambda, \mu, \nu \in \mathcal{P}(X) $, if $ \lambda \precsim \mu $ and $ \mu \precsim \nu $, then $ \lambda \precsim \nu $**Continuity**: for any $ \lambda, \mu, \nu \in \mathcal{P}(X) $, the sets $ \{ p \in [0, 1] : p \mu + (1-p) \nu \precsim \lambda \} $ and $ \{ p \in [0, 1] : \lambda \precsim p \mu + (1-p) \nu \} $ are closed in $[0, 1]$**Independence**: for any $ \lambda, \mu, \nu \in \mathcal{P}(X) $ and $ 0 < p \leq 1 $, we have $ \mu \precsim \nu $ if and only if $ p \mu + (1-p) \lambda \precsim p \nu + (1-p) \lambda $

A **von Neumann-Morgenstern utility** is a Borel random variable $ U: X \to \mathbb{R} $ such that for any $ \mu, \nu \in \mathcal{P}(X) $

$$ \mu \precsim \nu \iff \mathbb{E}_\mu[U] \leq \mathbb{E}_\nu[U] $$

The von Neumann-Morgenstern utility theorem asserts that a von Neumann-Morgenstern utility always exists on a finite set:

Let $X$ be a finite set, equipped with the discrete $\sigma$-algebra $2^X$. For any rational preference relation $\precsim$ on $\mathcal{P}(X)$, there exists a corresponding von Neumann-Morgenstern utility. Note that in this case, $\mathcal{P}(X)$ is just the standard $|X|-1$ dimensional simplex in $[0, 1]^{|X|}$.

**Question:**

Let $X$ be a Polish space and $\precsim$ a rational preference relation on $\mathcal{P}(X)$. What sufficient conditions can guarantee the existence of a corresponding von Neumann-Morgenstern utility? What about a **continuous** one?