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A nonlinear integral defined on partition of set and its fundamental properties

Nonlinear integrals play an important role in the information fusion. So far, many nonlinear integrals such as Sugeno integral, Choquet integral, pan-integral and Wang-integral have already been defined well and have been applied successfully to solve the problems of information fusion. All these existing nonlinear integrals of a function with respect to a set function are defined on a subset of a space. In many problems of information fusion such as decision tree generation in inductive learning, we often deal with the function defined on a partition of the space. Motivated by minimizing the classification information entropy of a partition while generating decision trees, this paper proposes a nonlinear integral of a function with respect to a non-negative set function on a partition. The basic properties of the proposed integral are discussed and the potential applications of the proposed integral to decision tree generation are outlined in this paper.