Stability of Wasserstein Distance Under Moment Convergence is a result from the foundations of optimal transport. For mathematical objects be a Polish mathematical objectsp 1mathematical objectsk,k,, Pp(X)mathematical objectsk Nmathematical objectsk objectsk objectsx0 X$, X d(x,x0)p\,dk(x) X d(x,x0)p\,d(x) and X d(x,x0)p\,. It helps organize the relationship between Monge maps, Kantorovich plans, duality, and Wasserstein geometry.