For pt.I see ibid., vol.13, no.8, p.2673-88 (1980). Concepts of classical and quantum global measurability introduced in an earlier paper are discussed in greater detail and in the context of an n-dimensional Riemannian manifold. The ideas are illustrated with diagrams and examples. A notion of exact measurability is introduced and is shown to imply quantum global measurability. The physically important Killing momenta are shown to be exactly, and hence quantum globally, measurable.