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. 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 (ΛX,⊑) such that X embeds as the subset of maximal elements of ΛX where the relative Lawson topology of ΛX induces the patch topology of X. racterise the probabilistic power domain of a stably locally compact space as a stochastically ordered space of probability measures.