  Can the joint measures of quenched disordered lattice spin models (with
finite range) on the product of spin-space and disorder-space be represented as
(suitably generalized) Gibbs measures of an ``annealed system''? - We prove
that there is always a potential (depending on both spin and disorder
variables) that converges absolutely on a set of full measure w.r.t. the joint
measure (``weak Gibbsianness''). This ``positive'' result is surprising when
contrasted with the results of a previous paper [K6], where we investigated the
measure of the set of discontinuity points of the conditional expectations
(investigation of ``a.s. Gibbsianness''). In particular we gave natural
``negative'' examples where this set is even of measure one (including the
random field Ising model). Further we discuss conditions giving the convergence
of vacuum potentials and conditions for the decay of the joint potential in
terms of the decay of the disorder average over certain quenched correlations.
We apply them to various examples. From this one typically expects the
existence of a potential that decays superpolynomially outside a set of measure
zero. Our proof uses a martingale argument that allows to cut (an infinite
volume analogue of) the quenched free energy into local pieces, along with
generalizations of Kozlov's constructions.
