Abstract
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 (∧ X, ⊆) 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.We characterise the probabilistic power domain of a stably locally compact space as a stochastically ordered space of probability measures.
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