K-拟加Sugeno积分刻画广义函数列的一致可积性

Uniform Integrability of Sequence of Generalized Functions Described by K-quasi Additive Sugeno Integral

K-拟加Sugeno积分是借助于诱导算子定义的一种新型非可加积分,它在广义积分理论和一些实际应用中发挥重要作用.为克服K-拟加测度不具有可加性的先天性不足,本文建立一类新的非可加积分模型“K-拟加Sugeno积分”,从而为进一步研究非可加积分理论开辟一个新途径.一方面,在K-拟加测度空间上通过诱导算子对广义可测函数定义了K-拟加Sugeno积分,并利用该积分的解析表示讨论了广义函数列的一致可积性和一致有界性.另一方面,在K-拟加测度空间上证明了非负广义函数列的一致有界性蕴含着一致可积性,进而在K-拟加Sugeno积分意义下给出了非负广义函数列一致可积的一个充要条件.

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.