The Scott topology induces the weak topology

Given a probability measure on a compact metric space, we construct an increasing chain of valuations on the upper space of the metric space whose least upper bound is the measure. We then obtain the expected value of any Holder continuous function with respect to the measure up to any precision. We prove that the Scott topology induces the weak topology of the space of probability measures in the following general setting: Whenever a separable metric space is embedded into a subset of the maximal elements of an ω-continuous dcpo, which is a G_δ subset of the dcpo equipped with the Scott topology, we show that the space of probability measures of the metric space equipped with the weak topology is then embedded into a subspace of the maximal elements of the probabilistic power domain of the dcpo. We present a novel application in the theory of periodic doubling route to chaos