Random sets arise as distinguished models to study probability under imprecision. This work aims to determine the connectedness of a random set in terms of its capacity functional (the probability of hitting a set). This problem is linked to the celebrated Choquet–Kendall–Matheron theorem, which states that a random closed set is characterized by its capacity functional. Hence, such functional must also characterize any topological property of a random set, such as its connectedness. While the capacity functional of a random closed set operates over the large class of compact sets, we show that the determination of connectedness can be checked only over a simple family of sets.

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关于 ℝ 的随机集的连通性

随机集作为杰出的模型出现，用于研究不精确下的概率。这项工作旨在根据容量泛函（命中集合的概率）来确定随机集合的连通性。这个问题与著名的 Choquet-Kendall-Matheron 定理有关，该定理指出随机闭集的特征在于其容量泛函。因此，这种泛函还必须刻画随机集的任何拓扑属性，例如它的连通性。虽然随机闭集的容量泛函在大类紧集上运行，但我们表明，连通性的确定只能在一个简单的集合族上进行检查。