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11.07.2019 | Methodologies and Application Open Access

# The $$\lambda$$-additive measure in a new light: the $$Q_{\nu }$$ measure and its connections with belief, probability, plausibility, rough sets, multi-attribute utility functions and fuzzy operators

Zeitschrift:
Soft Computing
Autoren:
József Dombi, Tamás Jónás
Wichtige Hinweise
Communicated by V. Loia.

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## Abstract

The aim of this paper is twofold. On the one hand, the $$\lambda$$-additive measure (Sugeno $$\lambda$$-measure) is revisited, and a state-of-the-art summary of its most important properties is provided. On the other hand, the so-called $$\nu$$-additive measure as an alternatively parameterized $$\lambda$$-additive measure is introduced. Here, the advantages of the $$\nu$$-additive measure are discussed, and it is demonstrated that these two measures are closely related to various areas of science. The motivation for introducing the $$\nu$$-additive measure lies in the fact that its parameter $$\nu \in (0,1)$$ has an important semantic meaning as it is the fix point of the complement operation. Here, by utilizing the $$\nu$$-additive measure, some well-known results concerning the $$\lambda$$-additive measure are put into a new light and rephrased in more advantageous forms. It is discussed here how the $$\nu$$-additive measure is connected with the belief-, probability- and plausibility measures. Next, it is also shown that two $$\nu$$-additive measures, with the parameters $$\nu _1$$ and $$\nu _2$$, are a dual pair of belief- and plausibility measures if and only if $$\nu _1+\nu _2 = 1$$. Furthermore, it is demonstrated how a $$\nu$$-additive measure (or a $$\lambda$$-additive measure) can be transformed to a probability measure and vice versa. Lastly, it is discussed here how the $$\nu$$-additive measures are connected with rough sets, multi-attribute utility functions and certain operators of fuzzy logic.

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