On monotone measures that provide the coincidence of the Choquet integral and the pan-integral

It is known that the (M)-property and the weak (M)-property of monotone measures were introduced in the studying the coincidence of the Choquet integral and the pan-integral. We proved that for a finite monotone measure, which is continuous from above, these two properties are equivalent. Besides that, some new properties and a criterion for additivity of monotone measures with the (M)-property and the weak (M)-property were presented. We also introduce the notion of the middle (M)-property and rise an open problem with the relationship between this property and the weak (M)-property.