When Scott is Weak on the Top

this paper, we solve the following two basic problems in constructive and computational mathematics. Suppose a probability measure on a compact metric space is given by its values on a countable base closed under finite unions and intersections. We construct an increasing chain of simple valuations on the upper space of the metric space, i.e. an approximating chain in the probabilistic power domain of the upper space, whose least upper bound is the probability measure. In order to state the second problem, we first recall a fundamental feature of the various domain-theoretic models for classical Hausdorff spaces, e.g. the Cantor domain \Sigm