Monotone Measures Defined by Pan-Integral - Advances in Pure Mathematics

APM > Vol.8 No.6, June 2018

Monotone Measures Defined by Pan-Integral ()

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Download as PDF (Size: 439KB) PP. 535-547

DOI: 10.4236/apm.2018.86031 698 Downloads 1,445 Views Citations

Author(s)

Peipei Wang¹, Minhao Yu², Jun Li^2*

Affiliation(s)

¹Property and Finance Department, National Museum of China, Beijing, China.
²School of Sciences, Communication University of China, Beijing, China.

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.

KEYWORDS

Monotone Measure, Pan-Integral, Sugeno Integral, Choquet Integral, Shilkret Integral

Share and Cite:

Wang, P. , Yu, M. and Li, J. (2018) Monotone Measures Defined by Pan-Integral. Advances in Pure Mathematics, 8, 535-547. doi: 10.4236/apm.2018.86031.

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