Monotone Measures Defined by Pan-Integral

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.