有关Hausdorff测度的两类覆盖形式  

Two forms of cover for Hausdorff measure

作　　者：丁丹[1] 郭晶[2] 盛中平[2] 

机构地区：[1]吉林建筑大学基础科学部,吉林长春130118 [2]东北师范大学数学与统计学院,吉林长春130024

出　　处：《东北师大学报（自然科学版）》2013年第4期1-5,共5页Journal of Northeast Normal University(Natural Science Edition)

基　　金：国家自然科学基金资助项目(11071035)

摘　　要：讨论了测度定义的具体覆盖形式,在普通球覆盖的基础上引入了广义球覆盖.利用两类球覆盖给出了Hausdorff测度定义的两种形式,并证明了在维数大于零时两种定义形式是等价的.指出了可数覆盖具有拒零性,Hausdorffδ-测度当δ为零时不存在,同时Lebesgue零维测度也没意义.计数测度在形式上不能统一到Lebesgue测度里,也不能统一到普通球覆盖下的Hausdorff测度里.只有在广义球覆盖下,才能形式上统一到Hausdorff测度中.同时,与普通球覆盖相比,利用广义球覆盖来定义测度,会使许多证明得以简化.This paper discusses the form of cover for measure, puts forward the generalized ball cover on the basis of the ordinary ball cover. Using the two forms of ball cover,it gives the two definitions on Hausdorff measure, and proves that the two definitions are equivalent supposing that dimension greater than zero. It points out that the countable cover has the zero-refusing property, and both the Hausdorff 0-measure and the Lebesgue zero-dimension measure all don＇t make sense. The counting measure is neither the Lebesgue measure nor the hausdorff measure with the ordinary ball cover. Only with the generalized ball cover,the counting measure is the Hausdorff measure. If the generalized ball cover is taken,many proofs can be simplified.

关 键 词：广义球覆盖 HAUSDORFF测度 计数测度 拒零性 

分 类 号：O174.12[理学—数学]