Spaces of Valuations

Abstract

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 V f X ⊂ V p X ⊂ VX . The main results of the paper are that V p X is the soberification of V f X , and that V p X 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 V p X and VX as dual spaces of certain spaces of real‐valued functions on X .