Atoms Cannot Be Split by a Measurable Map is a result from the foundations of optimal transport. For mathematical objects be a mathematical objects(Y, B)mathematical objectsT:(X, A) (Y, B)mathematical objectsA Amathematical objects objects0<(A)< objectsE Amathematical objectsE Amathematical objects(E)=0mat. It helps organize the relationship between Monge maps, Kantorovich plans, duality, and Wasserstein geometry.