  In this paper we define an orientation of a measured Gromov-Hausdorff limit
space of Riemannian manifolds with uniform Ricci bounds from below. This is the
first observation of orientability for metric measure spaces. Our orientability
has two fundamental properties. One of them is the stability with respect to
noncollapsed sequences. As a corollary we see that if the cross section of a
tangent cone of a noncollapsed limit space of orientable Riemannian manifolds
is smooth, then it is also orientable in the ordinary sense, which can be
regarded as a new obstruction for a given manifold to be the cross section of a
tangent cone. The other one is that there are only two choices for orientations
on a limit space. We also discuss relationships between $L^2$-convergence of
orientations and convergence of currents in metric spaces. In particular for a
noncollapsed sequence, we prove a compatibility between the intrinsic flat
convergence by Sormani-Wenger, the pointed flat convergence by Lang-Wenger, and
the Gromov-Hausdorff convergence, which is a generalization of a recent work by
Matveev-Portegies to the noncompact case. Moreover combining this compatibility
with the second property of our orientation gives an explicit formula for the
limit integral current by using an orientation on a limit space. Finally
dualities between de Rham cohomologies on an oriented limit space are proven.
