  The generalized $Q$-function of a spin system can be considered the outcome
probability distribution of a state subjected to a measurement represented by
the spin-coherent-state (SCS) positive-operator-valued measure (POVM). As
fundamental as the SCS POVM is to the 2-sphere phase-space representation of
spin systems, it has only recently been reported that the SCS POVM can be
performed for any spin system by continuous isotropic measurement of the three
total spin components [E. Shojaee, C. S. Jackson, C. A. Riofrio, A. Kalev, and
I. H. Deutsch, Phys. Rev. Lett. 121, 130404 (2018)]. This article develops the
theoretical details of the continuous isotropic measurement and places it
within the general context of curved-phase-space correspondences for quantum
systems. The analysis is in terms of the Kraus operators that develop over the
course of a continuous isotropic measurement. The Kraus operators of any spin
$j$ are shown to represent elements of the Lie group $\mathrm{SL}(2,{\mathbb
C})\cong\mathrm{Spin}(3,{\mathbb C})$, a complex version of the usual unitary
operators that represent elements of
$\mathrm{SU}(2)\cong\mathrm{Spin}(3,{\mathbb R})$. Consequently, the associated
POVM elements represent points in the symmetric space
$\mathrm{SU}(2)\backslash\mathrm{SL}(2,{\mathbb C})$, which can be recognized
as the 3-hyperboloid. Three equivalent stochastic techniques, (Wiener) path
integral, (Fokker-Planck) diffusion equation, and stochastic differential
equations, are applied to show that the continuous isotropic POVM quickly
limits to the SCS~\hbox{POVM}, placing spherical phase space at the boundary of
the fundamental Lie group $\mathrm{SL}(2,{\mathbb C})$ in an operationally
meaningful way. The Kraus-operator-centric analysis is representation
independent -- and therefore geometric (independent of any spectral information
about the spin components).
