  It is often desirable to summarise a probability measure on a space $X$ in
terms of a mode, or MAP estimator, i.e.\ a point of maximum probability. Such
points can be rigorously defined using masses of metric balls in the
small-radius limit. However, the theory is not entirely straightforward: the
literature contains multiple notions of mode and various examples of
pathological measures that have no mode in any sense. Since the masses of balls
induce natural orderings on the points of $X$, this article aims to shed light
on some of the problems in non-parametric MAP estimation by taking an
order-theoretic perspective, which appears to be a new one in the inverse
problems community. This point of view opens up attractive proof strategies
based upon the Cantor and Kuratowski intersection theorems; it also reveals
that many of the pathologies arise from the distinction between greatest and
maximal elements of an order, and from the existence of incomparable elements
of $X$, which we show can be dense in $X$, even for an absolutely continuous
measure on $X = \mathbb{R}$.
