A mode, or `most likely point', for a probability measure $\mu$ can be defined in various ways via the asymptotic behaviour of the $\mu$-mass of balls as their radius tends to zero. Such points are of intrinsic interest in the local theory of measures on metric spaces and also arise naturally in the study of Bayesian inverse problems and diffusion processes. Building upon special cases already proposed in the literature, this paper develops a systematic framework for defining modes through small-ball probabilities. We propose `common-sense' axioms that such definitions should obey, including appropriate treatment of discrete and absolutely continuous measures, as well as symmetry and invariance properties. We show that there are exactly ten such definitions consistent with these axioms, and that they are partially but not totally ordered in strength, forming a complete, distributive lattice. We also show how this classification simplifies for well-behaved $\mu$.

翻译：概率测度$\mu$的众数（或称“最可能点”）可通过$\mu$质量随球半径趋于零时的渐近行为以多种方式定义。这类点在度量空间测度局部理论中具有内在研究价值，同时也自然出现在贝叶斯反问题与扩散过程的研究中。本文在已有文献提出的特殊案例基础上，构建了通过小概率球定义众数的系统框架。我们提出了此类定义应遵循的“常识性”公理，包括对离散测度与绝对连续测度的恰当处理，以及对称性与不变性要求。我们证明与此类公理相容的定义恰好有十种，且它们在强度上构成部分而非全序关系，形成一个完备的分配格。我们还展示了该分类在$\mu$性质良好时的简化形式。