TITLE:
Monotone Measures Defined by Pan-Integral
AUTHORS:
Peipei Wang, Minhao Yu, Jun Li
KEYWORDS:
Monotone Measure, Pan-Integral, Sugeno Integral, Choquet Integral, Shilkret Integral
JOURNAL NAME:
Advances in Pure Mathematics,
Vol.8 No.6,
June
20,
2018
ABSTRACT: Given a pan-space and a nonnegative
measurable function f on measurable
space (X, A), the pan-integral of f with respect to monotone measure μ and
pan-operation determines a new
monotone measure on(X, A). Such the new monotone measure is absolutely
continuous with respect to the monotone measure μ. We show that the new monotone measure preserves some important
structural characteristics of the original monotone measures, such as
continuity from below, subadditivity, null-additivity, weak null-additivity and
(S) property. Since the pan-integral based on a pair of pan-operations covers the Sugeno
integral (based on ) and the Shilkret
integral (based on ), therefore, the previous related results for the Sugeno
integral are covered by the results presented here, in the meantime, some
special results related the Shilkret integral are also obtained.