Let $\{ x_n ,n = 1,2, \cdots \}$ 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 \[ M\left\{ {x_n \mid \mathfrak{F}} \right\} = \mathop \cup \limits_{y \in D} \left\{ {z:d\left( {z,y} \right) \leqq {\bf E}\left( {d\left( {x_n ,y} \right) | \mathfrak{F}} \right)} \right\}, \] 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.