Journal of the
Korean Mathematical Society JKMS

Types (over parameters) in the theory of atomless random variable structures correspond precisely to (conditional) distributions in probability theory. Moreover, the logic (resp.~metric) topology on the type space corresponds to the topology of weak (resp.~strong) convergence of distributions. In this paper, we study metrics between types. We show that type spaces under $d^*$-metric are isometric to Wasserstein spaces. Using optimal transport theory, two formulas for the metrics between types are given. Then, we give a new proof of an integral formula for the Wasserstein distance, and generalize some results in optimal transport theory.

Keywords: random variables, type spaces, Wasserstein distances

MSC numbers: 03C90, 60B10