The general Poincaré formula for λ-additive measures

Abstract

In this study, the general formula for λ-additive measure of union of n sets is introduced. Here, it is demonstrated that the well-known Poincaré formula of probability theory may be viewed as a limit case of our general formula. Moreover, it is also explained how this novel formula along with an alternatively parameterized λ-additive measure can be applied in theory of belief-and plausibility measures, and in theory of rough sets.

An alternative approach to quadratic scoring rules using continuous-valued logic

“…( 50) can be fitted to empirical data. Also, we plan to investigate how the weighting measure W could be modeled by λ-additive or ν-additive measures (see Dombi and Jónás (2019)). And, we plan to see whether the constructions of fuzzy affine models (see, e.g., Ji et al (2021Ji et al ( , 2022) can be adapted to quadratic scoring rules.…”

An Elementary Proof of the General Poincaré Formula for λ-additive Measures

“…[21]. Although there are many theoretical and applied articles that discuss the λadditive measure, the general form of λ-additive measure of the union of n sets has just recently been identified [4]. In [4], we proved that if X is a finite set, A 1 , .…”

“…where P(X) denotes the power set of X. Our proof in [4] is based on the fact that Q λ is representable [2]; that is, one has Q λ = h λ • µ for a uniquely determined additive measure µ : P(X) → [0, 1], where h λ : [0, 1] → [0, 1] is a strictly increasing bijection given via…”

“…[6,24,23,15,18,19]). Hence, our formula for the λ-additive measure of the union of n sets may play an important role in these areas of computer science [4].…”