Soft Computing https://doi.org/10.1007/s00500-019-04212-y METHODOLOGIES AND APPLICATION The -additive measure in a new light: the Q  measure and its connections with belief, probability, plausibility, rough sets, multi-attribute utility functions and fuzzy operators József Dombi 1 · Tamás Jónás 2 © The Author(s) 2019 Abstract The aim of this paper is twofold. On the one hand, the λ-additive measure (Sugeno λ-measure) is revisited, and a state- of-the-art summary of its most important properties is provided. On the other hand, the so-called ν -additive measure as an alternatively parameterized λ-additive measure is introduced. Here, the advantages of the ν -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 ν -additive measure lies in the fact that its parameter ν ∈ (0, 1) has an important semantic meaning as it is the fix point of the complement operation. Here, by utilizing the ν -additive measure, some well-known results concerning the λ-additive measure are put into a new light and rephrased in more advantageous forms. It is discussed here how the ν -additive measure is connected with the belief-, probability- and plausibility measures. Next, it is also shown that two ν -additive measures, with the parameters ν 1 and ν 2 , are a dual pair of belief- and plausibility measures if and only if ν 1 + ν 2 = 1. Furthermore, it is demonstrated how a ν -additive measure (or a λ-additive measure) can be transformed to a probability measure and vice versa. Lastly, it is discussed here how the ν -additive measures are connected with rough sets, multi-attribute utility functions and certain operators of fuzzy logic. Keywords Belief · Probability · Plausibility · λ-Additive measure · Rough sets · Multi-attribute utility functions 1 Introduction It is an acknowledged fact that the λ-additive measure (Sugeno λ-measure) (Sugeno 1974) is one of the most widely applied monotone measures (fuzzy measure). The useful- ness, versatility and applicability of λ-additive measures have inspired numerous theoretical and practical researches since Sugeno’s original results were published in 1974 (see, e.g., Magadum and Bapat 2018; Mohamed and Xiao 2003; Chi¸ tescu 2015; Chen et al. 2016; Singh 2018). Communicated by V. Loia. B Tamás Jónás jonas@gti.elte.hu József Dombi dombi@inf.u-szeged.hu 1 Institute of Informatics, University of Szeged, Szeged, Hungary 2 Institute of Business Economics, Eötvös Loránd University, Budapest, Hungary The aim of the present study is twofold. On the one hand, we will revisit the λ-additive measure and give a state-of-the- art summary of its most important properties. On the other hand, we will introduce the so-called ν -additive measure as an alternatively parameterized λ-additive measure, demon- strate the advantages of the ν -additive measure and point out that these two measures are closely related to various areas of science. The motivation for introducing the ν -additive mea- sure lies in the fact that its parameter ν ∈ (0, 1) has an important semantic meaning. Namely, ν is the fix point of the complement operation; that is, if the ν additive measure of a set has the value ν , then the ν -additive measure of its complement set has the value ν as well. It should be added that by utilizing the ν -additive measure, some well-known results concerning the λ-additive measure can be put into a new light and rephrased in more advantageous forms. Here, we will discuss how the ν -additive measure is connected with the belief-, probability- and plausibility measures (see, e.g., Wang and Klir 2013; Höhle 1987; Dubois and Prade 1980; Spohn 2012; Feng et al. 2014). Also, we will demonstrate that a ν -additive measure is a 123