Measurements and Gδ-subsets of domains

In this paper we study domains, Scott domains, and the existence of measurements. We use a space created by D. K. Burke to show that there is a Scott domain P for which max(P) is a G _δ-subset of P and yet no measurement μ on P has ker(μ) = max(P). We also correct a mistake in the literature asserting that [0, ω ₁) is a space of this type. We show that if P is a Scott domain and X ⊆ max(P) is a G _δ- subset of P, then X has a G _δ-diagonal and is weakly developable. We show that if X ⊆ max(P) is a G _δ-subset of P, where P is a domain but perhaps not a Scott domain, then X is domain-representable, first-countable, and is the union of dense, completely metrizable subspaces. We also show that there is a domain P such that max(P) is the usual space of countable ordinals and is a G _δ-subset of P in the Scott topology. Finally we show that the kernel of a measurement on a Scott domain can consistently be a normal, separable, non-metrizable Moore space.