Comparison of Decomposition Type Nonlinear Integrals Based on the Convergence Theorem

We consider several nonlinear integrals with respect to the monotone measures. We term the integrals defined by certain approximations via simple functions as decomposition type integrals. This category is classified based on the approximation directions (from above/below) and the disjointness of corresponding measurable set families (partition/covering). Furthermore, we add two classification points: finiteness of the summation (finite/countable) and sign of coefficients (non-negative/signed). This study aims to clarify the essential features of these integrals by considering several convergence theorems for each group of the decomposition type integrals.