关键词: 诱导算子, $K$-拟加 Sugeno 积分, 一致可积, 一致有界

Abstract: $K$-quasi additive Sugeno integral is a new non-additive integral defined by the induced operator, it plays an important role in the generalized integral theory and some practical applications. In order to overcome the inborn deficiency of $K$-quasi additive measure: without additivity, a new non-additive integral model ``$K$-quasi additive Sugeno integral" is introduced. This provides a new way to further study the theory of non-additive integral. On the one hand, on the $K$-quasi additive measure space, the $K$-quasi additive Sugeno integral with the generalized measurable function is defined by the induced operator, and the uniform integrability and uniform boundedness of sequence of generalized functions are discussed by using the analytic representation of the integral. On the other hand, on the $K$-quasi additive measure space, it is proved that the uniform boundedness of a sequence of nonnegative generalized functions contains uniformly integrability, and then a sufficient and necessary condition for the uniformly integrability of the sequence of nonnegative generalized functions is given in the sense of $K$-quasi additive Sugeno integral.

Key words: induced operator, $K$-quasi additive Sugeno integral, uniformly integrable, uniformly boundedness