Nonstandard probability theories have been developed for modeling random systems in complex spaces, such as, quantum systems. One of these theories, the MV-algebraic probability theory, involves the notions of state and observable, which were introduced by abstracting the properties of the Kolmogorovian probability measure and the classical random variable, as well as the notion of independence. Although within these nonstandard probability theories, many important theorems, including the strong law of large numbers (SLLN) for sequences of independent and identically distributed observables, have been considered, some practical applications require their further development. This paper is devoted to the development of the IVM-probability theory for the data described by interval-valued fuzzy random sets (IVM-events). The generalizations of Marcinkiewicz–Zygmund SLLN and Brunk–Prokhorov SLLN for independent IVM-events have been proved within this new theory. Our results open new possibilities in the theoretical analysis of imprecise random events in more complex spaces.

中文翻译：

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IVM 事件的强大数定律

已经开发了非标准概率理论来模拟复杂空间中的随机系统，例如量子系统。其中一种理论，即 MV 代数概率理论，涉及状态和可观察的概念，它们是通过抽象 Kolmogorovian 概率测度和经典随机变量的性质以及独立性概念引入的。尽管在这些非标准概率理论中，已经考虑了许多重要的定理，包括独立同分布的可观测序列的强大数定律 (SLLN)，但一些实际应用需要进一步发展。本文致力于为区间值模糊随机集 (IVM-events) 描述的数据开发 IVM 概率理论。Marcinkiewicz-Zygmund SLLN 和 Brunk-Prokhorov SLLN 对独立 IVM 事件的推广已经在这个新理论中得到证明。我们的结果为更复杂空间中不精确随机事件的理论分析开辟了新的可能性。