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

Ultralimits of Wasserstein spaces and metric measure spaces with Ricci curvature bounded from below