It is often desirable to summarize probability measures in the space $X$ in terms of modal or MAP estimators, or maximum probabilities. Such points can be precisely defined using the mass of a metric ball with small radius limits. However, the theory is not entirely simple. The literature contains multiple concepts of modes and various examples of pathological measurements that do not have modes in any sense. Since the mass of the ball induces a natural ordering in terms of X, this article aims to shed light on some of the problems of nonparametric MAP estimation in terms of order theory. Inverse problem community. This perspective opens up an attractive proof strategy based on the Cantor-Klatovsky intersection theorem. It also reveals the distinction between maximal and maximal elements of order, and many of the pathologies arising from the presence of unique elements of X$. with $X = \mathbb{R}$ .
