We show, by a simple and direct proof, that if a bounded valuation on a directed complete partial order (dcpo) is the supremum of a directed family of simple valuations then it has a unique extension to a measure on the Borel oe-algebra of the dcpo with the Scott topology. It follows that every bounded and continuous valuation on a continuous domain can be extended uniquely to a Borel measure. The result also holds for oe-finite valuations, but fails for dcpo's in general. 1

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After surveying recent work and new techniques in domain theoretic models of spaces, we introduce a new topological concept called recurrence, and consider some of its applications to the model problem.

In this paper, we survey the use of order-theoretic topology in theoretical computer science, with an emphasis on applications of domain theory. Our focus is on the uses of order-theoretic topology in programming language semantics, and on problems of potential interest to topologists that stem from concerns that semantics generates. Keywords: Domain theory, Scott topology, power domains, untyped lambda calculus Subject Classification: 06B35,06F30,18B30,68N15,68Q55 1 Introduction Topology has proved to be an essential tool for certain aspects of theoretical computer science. Conversely, the problems that arise in the computational setting have provided new and interesting stimuli for topology. These problems also have increased the interaction between topology and related areas of mathematics such as order theory and topological algebra. In this paper, we outline some of these interactions between topology and theoretical computer science, focusing on those aspects that have been mo...

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.

Abstract. We discuss some of the choices that arise when one tries to make the idea of physical determinism more precise. Broadly speaking, ‘ontological ’ notions of determinism are parameterized by one’s choice of mathematical ideology, whilst ‘epistemological ’ notions of determinism are parameterized by the choice of an appropriate notion of computability. We present some simple examples to show that these choices can indeed make a difference to whether a given physical theory is ‘deterministic’ or not. Keywords: Laplace’s demon, physical determinism, philosophy of mathematics, notions of computability. 1