Atoms of weakly null-additive monotone measures and integrals

Keywords: Monotone measure Atom Weak null-additivity Regularity Sugeno integral Choquet integral a b s t r a c t In this paper, we prove some properties of atoms of weakly null-additive monotone measures. By using the regularity and weak null-additivity, a sin-gleton characterization of atoms of monotone measures on a metric space is shown. It is a generalization of previous results obtained by Pap. The calculation of the Sugeno integral and the Choquet integral over an atom is also presented, respectively. Similar results for recently introduced universal integral are also given. Following these results, it is shown that the Sugeno integral and the Choquet integral over an atom of monotone measure is maxitive linear and standard linear, respectively. Convergence theorems for the Sugeno integral and the Choquet integral over an atom of a monotone measure are also shown. An atom of a measure is an important concept in the classical measure theory [6] and probability theory. This concept was generalized in non-additive measure theory. The atoms for submeasures on locally compact Hausdorff spaces were discussed by Dobrakov [4]. In 1991, Suzuki [24] first introduced the concept of an atom of fuzzy measures (non-negative monotone set functions with continuity from below and above and vanishing at £), and investigated some analytical properties of atoms of fuzzy measures. Pap showed a singleton characterization of atoms of regular null-additive monotone set functions, i.e., if a non-negative monotone set function l is regular and null-additive, then every atom of l has an outstanding property that all the mass of the atom is concentrated on a single point in the atom. In this paper, we shall further investigate some properties of atoms of weakly null-additive monotone measures on metric spaces. We shall show that if a regular monotone measure is weakly null-additive, then the previous results obtained by Pap [21] remain valid. This fact makes easy the calculation of the Sugeno integral and the Choquet integral over an atom of a monotone measure which is regular and countably weakly null-additive. Following these results, it is shown that the Sugeno integral and the Choquet integral over an atom of a monotone measure is maxitive linear and standard linear (cf.[17]),