Spaces of Valuations

Reinhold Heckmann


Valuations are measure-like 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 Vf X C Vp X C V X. The main results of the paper are that Vp X is the soberification of Vf X, and that Vp 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 Vp X and V X as dual spaces of certain spaces of real-valued functions on X.

Key words: Topological spaces of valuations, topological cones, integration, soberification

AMS classification codes: 54B99, 28A25, 46A99, 46E99, 54D35

[ (23p, 92k, reformatted)]

Reinhold Heckmann /