Spaces of Valuations

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Abstract
Valuations are measurelike functions mapping the open sets of a topological space X into positive real numbers. They can be classified into finite, point continuous, and Scott continuous valuations. We define corresponding spaces of valuations VfX⊂ VpX⊂VX. The main results of the paper are that VpX is the soberification of VfX, and that VpX is the free sober locally convex topological cone over X. From this universal property, the notion of the integral of a real‐valued function over a Scott continuous valuation can be easily derived. The integral is used to characterize the spaces VpX and VX as dual spaces of certain spaces of real‐valued functions on X.

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