Subsethood, which is to measure the degree of set inclusion relation, is predominant in fuzzy set theory. This paper introduces some basic concepts of spatial granules, coarse-fine relation, and operations like meet, join, quotient meet and quotient join. All the atomic granules can be hierarchized by set-inclusion relation and all the granules can be hierarchized by coarse-fine relation. Viewing an information system from the micro and the macro perspectives, we can get a micro knowledge space and a micro knowledge space, from which a rough set model and a spatial rough granule model are respectively obtained. The classical rough set model is the special case of the rough set model induced from the micro knowledge space, while the spatial rough granule model will be play a pivotal role in the problem-solving of structures. We discuss twelve axioms of monotone increasing subsethood and twelve corresponding axioms of monotone decreasing supsethood, and generalize subsethood and supsethood to conditional granularity and conditional fineness respectively. We develop five conditional granularity measures and five conditional fineness measures and prove that each conditional granularity or fineness measure satisfies its corresponding twelve axioms although its subsethood or supsethood measure only hold one of the two boundary conditions. We further define five conditional granularity entropies and five conditional fineness entropies respectively, and each entropy only satisfies part of the boundary conditions but all the ten monotone conditions.

翻译：子集测度用于衡量集合包含关系的程度，在模糊集理论中占据主导地位。本文介绍了空间粒子的基本概念、粗细关系以及交、并、商交、商并等运算。所有原子粒子可通过集合包含关系分层，所有粒子可通过粗细关系分层。从微观与宏观视角审视信息系统，可分别得到微观知识空间与宏观知识空间，并由此分别建立粗糙集模型和空间粗糙粒子模型。经典粗糙集模型是微观知识空间诱导的粗糙集模型的特例，而空间粗糙粒子模型将在结构问题求解中发挥关键作用。本文讨论了单调递增子集测度的十二个公理及其对应的十二个单调递减超集测度公理，并将子集测度与超集测度分别推广为条件粒度与条件精细度。我们开发了五种条件粒度测度和五种条件精细度测度，并证明每个条件粒度或精细度测度均满足其对应的十二个公理，尽管其子集测度或超集测度仅满足两个边界条件之一。进一步，我们分别定义了五种条件粒度熵和五种条件精细度熵，每个熵仅满足部分边界条件，但均满足全部十个单调条件。