  We study the mathematical structure of the notion of measurement space, which
extends aspects of noncommutative topology that are based on quantale theory.
This yields a geometric model of physical measurements that provides a realist
picture, yet also operational, such that measurements and classical information
arise interdependently as primitive concepts. A derived notion of classical
observer caters for a mathematical formulation of Bohr's classical/quantum
divide. Two important classes of measurement spaces are obtained, respectively
from C*-algebras and from second-countable locally compact open sober
topological groupoids. The latter yield measurements of classical type and
relate to Schwinger's notion of selective measurement. We show that the
measurement space associated to the reduced C*-algebra of any second-countable
locally compact Hausdorff \'etale groupoid is canonically equipped with a
classical observer, and we establish a correspondence between properties of the
observer and properties of the groupoid.
