  We investigate the stability of the Wasserstein distance, a metric structure
on the space of probability measures arising from the theory of optimal
transport, under metric ultralimits. We first show that if
$(X_{i},d_{i})_{i\in\mathbb{N}}$ is a sequence of metric spaces with metric
ultralimit $(\hat{X},\hat{d})$, then the p-Wasserstein space
$(\mathcal{P}_{p}(\hat{X}),W_{p})$ embeds isometrically in a canonical fashion
into the metric ultralimit of the sequence of p-Wasserstein spaces
$(\mathcal{P}_{p}(X_{i}),W_{p})$. Second, using a notion of ultralimit of
metric measure spaces modeled on the one introduced by Elek, we use the
machinery of ultralimits of Wasserstein spaces to prove that an ultralimit of
CD(K,$\infty$) spaces is a CD(K,$\infty$) space. This provides a new proof that
the CD(K,$\infty$) property is stable under pointed measured Gromov
convergence. Along the way, we establish some basic results on how the Loeb
measure construction interacts with Wasserstein distances as well as integral
functionals, which may be of independent interest.
