A Reeb graph is a graphical representation of a scalar function on a topological space that encodes the topology of the level sets. A Reeb space is a generalization of the Reeb graph to a multiparameter function. 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 introduce a Reeb graph with local smoothing and prove its stability with respect to the interleaving distance. We then prove the stability of a Reeb graph of a metric measure space with respect to the measure, defined using the distance to a measure or the kernel distance to a measure, respectively.

翻译：Reeb图是对拓扑空间上标量函数的图形化表示，用于编码水平集的拓扑结构。Reeb空间是Reeb图在多参数函数情形下的推广。本文提出了一种融合测度概念的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，我们引入了测度论Reeb图与Reeb空间。主要目标是在考虑测度分布的同时，增强Reeb图与Reeb空间表征标量场拓扑特征的鲁棒性。我们首先引入带局部平滑的Reeb图，并证明其关于交错距离的稳定性。随后，分别利用到测度的距离或到测度的核距离，证明了度量测度空间Reeb图关于测度的稳定性。