A Reeb graph is a graphical representation of a scalar function $f: X \to \mathbb{R}$ on a topological space $X$ that encodes the topology of the level sets. A Reeb space is a generalization of the Reeb graph to a multivariate function $f: X \to \mathbb{R}^d$. In this paper, we propose novel constructions of Reeb graphs and Reeb spaces that incorporate the use of a measure. Specifically, we introduce measure theoretic Reeb graphs and Reeb spaces when the domain or the range is modeled as a metric measure space (i.e.,~a metric space equipped with a measure). Our main goal is to enhance the robustness of the Reeb graph and Reeb space in representing the topological features of a scalar field while accounting for the distribution of the measure. We first prove the stability of our measure theoretic constructions with respect to the interleaving distance. We then prove their stability with respect to the measure, defined using the distance to a measure or the kernel distance to a measure, respectively.

翻译：Reeb图是标量函数$f: X \to \mathbb{R}$在拓扑空间$X$上的图形表示，用于编码水平集的拓扑结构。Reeb空间则是将Reeb图推广至多变量函数$f: X \to \mathbb{R}^d$的情形。本文提出融入测度的Reeb图与Reeb空间的新构造方法。具体而言，当定义域或值域被建模为度量测度空间（即配备测度的度量空间）时，我们引入基于测度的Reeb图与Reeb空间。主要目标是在兼顾测度分布的同时，增强Reeb图与Reeb空间表征标量场拓扑特征的鲁棒性。我们首先证明了基于测度构造关于交错距离的稳定性，继而分别证明了其关于测度（由到测度的距离或到测度的核距离定义）的稳定性。