  We present a general duality result for Wasserstein distributionally robust
optimization that holds for any Kantorovich transport cost, measurable loss
function, and nominal probability distribution. Assuming an interchangeability
principle inherent in existing duality results, our proof only uses
one-dimensional convex analysis. Furthermore, we demonstrate that the
interchangeability principle holds if and only if certain measurable projection
and weak measurable selection conditions are satisfied. To illustrate the
broader applicability of our approach, we provide a rigorous treatment of
duality results in distributionally robust Markov decision processes and
distributionally robust multistage stochastic programming. Additionally, we
extend our analysis to other problems such as infinity-Wasserstein
distributionally robust optimization, risk-averse optimization, and globalized
distributionally robust counterpart.
