Extension of valuations on locally compact sober spaces

We show that every locally finite continuous valuation defined on the lattice of open sets of a regular or locally compact sober space extends uniquely to a Borel measure. In the sequel we derive a maximal point space representation for any locally compact sober space (X, G). That is, we show that there exists a continuous poset (LambdaX, subset of or equal to) such that X embeds as the subset of maximal elements of LambdaX where the relative Lawson topology of LambdaX induces the patch topology of X. We characterise the probabilistic power domain of a stably locally compact space as a stochastically ordered space of probability measures. (C) 2001 Elsevier Science B.V. All rights reserved.