  We investigate two aspects of the elementary example of POVMs on the
Euclidean plane, namely their status as quantum observables and their role as
quantizers in the integral quantization procedure. The compatibility of POVMs
in the ensuing quantum formalism is discussed, and a Naimark dilation is found
for the quantum operators. The relation with Toeplitz quantization is
explained. A physical situation is discussed, where we describe the linear
polarization of the light with the use of Stokes parameters. In particular, the
case of sequential measurements in a real bidimensional Hilbert space is
addressed. An interpretation of the Stokes parameters in the framework of
unsharp or fuzzy observables is given. Finally, a necessary condition for the
compatibility of two dichotomic fuzzy observables which provides a condition
for the approximate joint measurement of two incompatible sharp observables is
found.
