关于Poincaré氏注记和定义在*(Rn)上的一种非标准测度

On Poincaré′s Remark and a Kind ofNonstandard Measure Defined on *(Rn)

英文关键词:Poincaré's remark  Cantor's continuum  enlargement  hyperfinite set  hyperstandard measure.

中文摘要:

      受到Poincaré氏评注的启发，本文引入了一种新的非标准测度，简称为μ_h测度.文中应用非标准分析方法证明了标准实数集R在^*R上的μ_h测度为零.文末还指出了μ_h测度与Lebesgue测度具有相异本质的原由.

英文摘要:

      Here concerned is a certain kind of non-standard measure defined on the n-dimensional Euclidean space ^*(Rⁿ), which (with n = 1) can be used to show that any standard linear point-set or the usual ordered field R of real numbers is of measure zero. The proposition just mentioned is basically consistent with Poincaré's famous remark which renders a deep insight into the paradoxical structural nature of Cantor's continuum consisting precisely of all distinct real numbers.