On the equivalence of the Choquet integral and the pan-integrals from above

摘要

A monotone measure μ is said to have dual (M)-property if for any A ⊂ B, there exists C with A ⊂ C ⊂ B such that μ(C)=μ(A)andμ(B)=μ(C)+μ(B∖C). By using this concept, we study the relationship of the Choquet integral and the pan-integral from above. We prove that the Choquet integral coincides with the pan-integral from above if μ has dual (M)-property. When the underlying space is finite, we prove that the dual (M)-property is also necessary for the coincidence of these two integrals. Thus we provide a necessary and sufficient condition for the equivalence of the Choquet integral and the pan-integral from above on a finite space.