取值于R^d空间上函数积分的非标准定义

Integration of R^d space valued functions defined by nonstandard analysis

目的用非标准分析的方法给出取值于Rd空间上函数积分的定义。方法在非标准扩大模型下,利用转换原理,通过给出取值于Rd空间上阶梯函数积分的非标准定义。结果用非标准分析的方法给出了取值于Rd空间上函数积分的非标准定义。在此基础上,给出取值于Rd空间上函数积分的另一种形式及其性质。结论取值于Rd空间上函数的积分与取值于Banach空间函数的积分具有相同的性质,充分体现了非标准分析方法简洁直观的特点。

Aim To define the integration of Rd space valued functions by nonstandard analysis. Methods In the nonstandard enlargement model,used transfer principle,the integration of Rd space valued step function is defined by nonstandard analysis. Results The nonstandard definition of integration of Rd space valued function is given,and based on it,another form of nonstandard definition of integration of Rd space valued function and some properties of integration of Rd space valued function are given. Conclusion The properties of integration of Rd space val-ued function are consistent with the properties of integration of Banach space valued function,so the advantages of nonstandard analysis is showed.