Martingales on Metric Spaces

Let $\{x_n,n=1,2,\dots\}$ be a random sequence with values in a compact metric space $X$. Following Doss, we define the conditional mathematical expectation of $x_n$ with respect to the Borel field $\mathfrak{F}$ as the (random) set
$$M\left\{{x_n|\mathfrak{F}}\right\}=\mathop\cup\limits_{y\in D}\left\{{z:d\left({z,y}\right)\leq{\mathbf E}\left({d\left({x_n,y}\right)|\mathfrak{F}}\right)}\right\},$$
where $d(\cdot,\cdot)$ is the metric and $D$ is a countable dense subset of $X$. Let $\mathfrak{F}_n$ be an increasing sequence of Borel fields, such that $x_n$ is $\mathfrak{F}_n$-measurable. The process $x_n$ is called a (generalized) martingale if $x_n\in M\{x_{n+1}| \mathfrak{F}_n\}$ with probability one.