A Philosophical Foundation of Non-Additive Measure and Probability |
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Authors: | Sebastian Maaß |
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Institution: | 1. Departement Mathematik, ETH Zürich, R?mistrasse 101, 8092, Zürich, Switzerland
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Abstract: | In this paper, non-additivity of a set function is interpreted as a method to express relations between sets which are not
modeled in a set theoretic way. Drawing upon a concept called “quasi-analysis” of the philosopher Rudolf Carnap, we introduce
a transform for sets, functions, and set functions to formalize this idea. Any image-set under this transform can be interpreted
as a class of (quasi-)components or (quasi-)properties representing the original set. We show that non-additive set functions
can be represented as signed σ-additive measures defined on sets of quasi-components. We then use this interpretation to justify
the use of non-additive set functions in various applications like for instance multi criteria decision making and cooperative
game theory. Additionally, we show exemplarily by means of independence, conditioning, and products how concepts from classical
measure and probability theory can be transfered to the non-additive theory via the transform. |
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Keywords: | conditioning independence M?bius transform non-additive measure products quasi-analysis |
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