  We define a new two-parameter family of metrics on a subspace of Borel
probability measures on a metric fiber bundle, called the disintegrated
Monge--Kantorovich metrics. We then prove the disintegrated Monge--Kantorovich
metrics are complete, separable (except an endpoint case), geodesic spaces,
with a dual representation. Additionally, we prove existence and duality for an
associated barycenter problem, and provide conditions for uniqueness of a
barycenter. These results on barycenter problems for the disintegrated
Monge--Kantorovich metrics also yield the corresponding existence, duality, and
uniqueness results for classical Monge--Kantorovich barycenters in a wide
variety of spaces, including a uniqueness result on any connected, complete
Riemannian manifold, with or without boundary, with no restriction on the
geometry of any kind.
