On Disintegration of Measures in Optimal Transport

As part of a prelude to the applications of optimal transport theory to econometrics and machine learning, this tutorial paper focus on the notion of disintegration in measure theory in the analysis of Wassersrein metrics which are useful, in particular, for machine learning and financial risks. An elementary exposition on using disintegration to prove that Waserstein divergence is a bona fide metric is provided.