Inner and outer representation by Choquet integral

The representations of comonotonically additive and monotone functionals are investigated. An i-regular non-additive measure is defined as a non-additive measure whose values can be calculated from its values on the inner compact sets and the values on compact sets can be calculated from its values on the outer open sets. The basic properties of an i-regular non-additive measure are shown. The inner representation of the functional is proposed and is compared with the outer representation.