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Uncertainty Extension Theorem and Product Uncertain Measure

The additivity axiom of classical measure theory has been challenged by many mathematicians. Different replacements of the additivity correspond with different theory. In uncertainty theory, the additivity is replaced with self-duality and countable subadditivity. Similar to classical measure theory, there are some properties studied in uncertainty theory. Given the measure of each singleton set, the measure can be fully and uniquely determined in the sense of the maximum uncertainty principle. Generally speaking, a product uncertain measure may be defined in many ways, in this paper, a kind of definition is proposed.