Relationship between the concave integrals and the pan-integrals on finite spaces

This study discusses the relationship between the concave integrals and the pan-integrals on finite spaces. The minimal atom of a monotone measure is introduced and some properties are investigated. By means of two important structure characteristics related to minimal atoms: minimal atoms disjointness property and subadditivity for minimal atoms, a necessary and sufficient condition is given that the concave integral coincides with the pan-integral with respect to the standard arithmetic operations + and ⋅ on finite spaces. Following this result, we have shown that these two integrals coincide if the underlying monotone measure is subadditive.