Fractals and domain theory

Abstract

We show that a measurement μ on a continuous dcpo D extends to a measurement μ on the convex powerdomain CD iff it is a Lebesgue measurement. In particular, ker μ must be metrizable in its relative Scott topology. Moreover, the space ker μ in its relative Scott topology is homeomorphic to the Vietoris hyperspace of ker μ, i.e., the space of nonempty compact subsets of ker μ in its Vietoris topology – the topology induced by any Hausdorff metric. This enables one to show that Hutchinson's theorem holds for any finite set of contractions on a domain with a Lebesgue measurement. Finally, after resolving the existence question for Lebesgue measurements on countably based domains, we uncover the following relationship between classical analysis and domain theory: For an ω -continuous dcpo D with max(D) regular, the Vietoris hyperspace of max(D) embeds in max(CD) as the kernel of a measurement on CD.