Generalized Measure Theory

Abstract

IFSR was established ''to stimulate all activities associated with the scientific study of systems and to coordinate such activities at international level.'' The aim of this series is to stimulate publication of high-quality monographs and textbooks on various topics of systems science and engineering. This series complements the Federation's other publications.A Continuation Order Plan is available for this series. A continuation order will bring delivery of each new volume immediately upon publication. Volumes are billed only upon actual shipment. For further information please contact the publisher.Volumes 1-6 were published by Pergamon Press.

Coincidences of the Concave Integral and the Pan-Integral

“…In [13] (see also [9,27,28]) the relationship between the the concave integral and the Choquet integral was discussed, as follows: The following results were shown in [14] (Theorems 10.7 and 10.8 in [14]). Proof.…”

“…Note that µ is then a belief measure [14] which is k-additive [30]. Clearly, µ satisfies the constraints of Theorem 11, and thus Cav µ = Pan µ .…”

“…When µ is a monotone measure, the triple (X, A, µ) is called a monotone measure space ( [14,17,18]). In some literature, such a monotone measure µ constrained by the boundary condition µ(X) = 1 is also called a capacity or a fuzzy measure or a nonadditive probability, etc.…”

“…The concept of a pan-integral [8,14] involves two binary operations, the pan-addition ⊕ and pan-multiplication ⊗ of real numbers (see also [14,18,[22][23][24][25][26]). In this paper we only consider the pan-integrals with respect to the usual addition + and usual multiplication ·.…”

“…In [14] the order relationship between the pan-integral (with respect to the usual addition + and usual multiplication ·) and the Choquet integral was shown by using the subadditivity and superadditivity of monotone measures.…”