The Kantorovich norm, also known as the Wasserstein metric, quantifies the distance between probability measures by considering the optimal transport problem. It involves integrating distances between measures to determine how one measure can be transformed into another with minimal cost. The pushforward measure is a key concept in this context, as it describes how a measure is transformed under a function. Understanding the Kantorovich norm is essential for applications in fields like statistics, economics, and machine learning. This metric provides a rigorous framework for comparing distributions in a meaningful way.