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 ${{G}_{\delta }}$-subset of $P$ and yet no measurement $\mu $ on $P$ has $\text{ker(}\mu \text{)}\,=\,\max (P)$. We also correct a mistake in the literature asserting that $[0,\,{{\omega }_{1}})$ is a space of this type. We show that if $P$ is a Scott domain and $X\,\subseteq \,\max (P)$ is a ${{G}_{\delta }}$-subset of $P$, then $X$ has a ${{G}_{\delta }}$-diagonal and is weakly developable. We show that if $X\,\subseteq \,\max (P)$ is a ${{G}_{\delta }}$-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 ${{G}_{\delta }}$-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.

van Mill et al. posed in ‘Classes defined by stars and neighborhood assignments’, Topology Appl.154 (2007), 2127–2134 the following question: Is a star-compact space metrizable if it has a Gδ-diagonal? In this paper, we give a negative answer to this question.