  We study the mathematical structure of covariant phase observables. Such an
observable can alternatively be expressed as a phase matrix, as a sequence of
unit vectors, as a sequence of phase states, or as an equivalent class of
covariant trace-preserving operations. Covariant generalized operator measures
are defined by structure matrices which form a W*-algebra with phase matrices
as its subset. The properties of the Radon-Nikodym derivatives of phase
probability measures are studied.
