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$P$ for which \max (P)$\max (P)$ is a {{G}_{\delta }}${{G}_{\delta }}$-subset of P$P$ and yet no measurement \mu $\mu$ on P$P$ has \text{ker(}\mu \text{)}\,=\,\max (P)$\text{ker(}\mu \text{)}\,=\,\max (P)$. We also correct a mistake in the literature asserting that [0,\,{{\omega }_{1}})$[0,\,{{\omega }_{1}})$ is a space of this type. We show that if P$P$ is a Scott domain and X\,\subseteq \,\max (P)$X\,\subseteq \,\max (P)$ is a {{G}_{\delta }}${{G}_{\delta }}$-subset of P$P$, then X$X$ has a {{G}_{\delta }}${{G}_{\delta }}$-diagonal and is weakly developable. We show that if X\,\subseteq \,\max (P)$X\,\subseteq \,\max (P)$ is a {{G}_{\delta }}${{G}_{\delta }}$-subset of P$P$, where P$P$ is a domain but perhaps not a Scott domain, then X$X$ is domain-representable, first-countable, and is the union of dense, completely metrizable subspaces. We also show that there is a domain P$P$ such that \max (P)$\max (P)$ is the usual space of countable ordinals and is a {{G}_{\delta }}${{G}_{\delta }}$-subset of P$P$ 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.