P-可测函数的构造性质  

The Structural Properties of P-measurable Function

作　　者：赵占平[1] 周厚勇[2] 史开泉[3] 

机构地区：[1]黄淮学院经济管理系,河南驻马店463000 [2]黄淮学院数学科学系,河南驻马店463000 [3]山东大学数学与系统科学学院,山东济南250100

出　　处：《数学的实践与认识》2015年第24期284-290,共7页Mathematics in Practice and Theory

基　　金：国家自然科学基金(11371164);河南省基础与前沿技术研究计划项目(122300410071)

摘　　要：可测函数的构造性质是定义它关于测度μ的积分的理论基础.为了在P-测度空间上定义P-积分,借鉴可测函数的构造性质,引入了P-示性函数、P-简单函数、P-初等函数以及P-可测函数的概念,在此基础上系统地研究了P-实可测函数、有界P-实可测函数和非负P-可测函数与P-简单函数序列及P-初等函数序列的收敛关系;找出了P-实可测的充分必要条件;证明了实P-可测函数正部和负部都是非负P-实可测函数,最终得出任何P-实可测函数均可以表示为二非负P-可测函数之差,为定义P-积分提供了理论依据.The Structural properties of measurable function are theoretical basis of the P- integral on defining it about measure ~. In order to define the P-integral in the P-measure space, By referencing the structural properties of measurable function, and by introducing the concepts of P-indicative function, P-simple function, P-elementary function and P-measurable function, We systematically study the Convergence relation of the P^real measurable function, bounded P-real measurable function, non-negative P-measurable function, P-simple function sequence and P-elementary function sequence. Further, we find out the sufficient and necessary conditions of P-real measurable and prove that the positive part and negative part of P- real measurable function are non-negative P-real measurable function. Finally we obtain that any P- real measurable function can expressed as the differecce of two non-negative measurable function, which provides a theoretical basis for the definition of the P-integral.

关 键 词：P-简单函数 P-初等函数 P-可测函数 构造性质 

分 类 号：O174.1[理学—数学]