  The concept of metastate measures on the states of a random spin system was
introduced to be able to treat the large-volume asymptotics for complex
quenched random systems, like spin glasses, which may exhibit chaotic volume
dependence in the strong-coupling regime. We consider the general issue of the
extremal decomposition for Gibbsian specifications which depend measurably on a
parameter that may describe a whole random environment in the infinite volume.
Given a random Gibbs measure, as a measurable map from the environment space,
we prove measurability of its decomposition measure on pure states at fixed
environment, with respect to the environment. As a general corollary we obtain
that, for any metastate, there is an associated decomposition metastate, which
is supported on the extremes for almost all environments, and which has the
same barycenter.
