We construct a homeomorphism between the compact regular locale of integrals on a Riesz space and the locale of measures(valuations) on its spectrum. In fact, we construct two geometric theories and show that they are biinterpretable. The constructions are elementary and tightly connected to the Riesz space structure. 2000 Mathematics Subject Classification 06D22, 28C05
Vickers @cite_19 presents another variant of the Riesz representation theorem. His construction works for locales which are not necessarily compact completely regular. However, his integrals have their values in the lower (or upper) reals, as opposed to the Dedekind reals. A locale of valuations was first presented by Heckman @cite_7 .
@cite_19 @cite_7
2144063313 1565369011
An account of lower and upper integration is given. It is constructive in the sense of geometric logic. If the integrand takes its values in the nonnegative lower reals, then its lower integral with respect to a valuation is a lower real. If the integrand takes its values in the non-negative upper reals, then its upper integral with respect to a covaluation and with domain of integration bounded by a compact subspace is an upper real. Spaces of valuations and of covaluations are deflned. Riemann and Choquet integrals can be calculated in terms of these lower and upper integrals. The probabilistic power domain construction of Jones and Plotkin [6, 7] is defined by a construction on dcpo's. We present alternative definitions in terms of information systems a la Vickers [12], and in terms of locales. On continuous domains, all three definitions coincide.