Abstract
This paper has two major contributions: (1) Shafer's belief functions are extended from finite sets to general universes (sets). (2) Belief function theory on a universe Ω with countable focal elements is interpreted as a probability theory on the product space Ω × [0,1] A counter intuitive point of a belief function is its ignoring the impact of the intersections among focal elements. One obvious intuitive rational is: The basic probability assignment (bpa) is a measurement of some disjoint "information" on focal elements. Item #2 show that this disjoint "information" is a partition (equivalence relation) on the product space. Here are interesting corollary and new exploration: (3) Belief function is an inner probability, if all focal elements are disjoint. (4) Issues for Bel of uncountable focal elements are lightly touched.
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