浅析n维欧氏空间上Borel集的构造

A Brief Analysis on the Construction of Borel Sets in n Dimensional Euclidean Space

针对n维欧氏空间上Borel集的构造问题,提出几个具有测度论特色的结果加以详细讨论.利用n维欧氏空间中左端点形如mi/2~l(其中mi为整数,l为正整数),且长度均为1/2~l的那些左开右闭区间形成的集类A_l的优良结构,结合实数域上的区间划分、不等式与拓扑技巧,证明了A_l是n维欧氏空间的可数无限划分,且随着l变得越大A_l变得越精细,对n维欧氏空间中开集中的任意一点来说,当l充分大时,A_l中包含该点的那个成员必定包含于该开集中;在此基础上用反证法证明了n维欧氏空间中任一开集都可表示成至多可数无限多个两两不交的n维左开右闭区间之并;最后以此结论为工具,介绍了n维欧氏空间上Borel代数的几个较小生成元.

Focusing on the construction of Borel sets in n dimensional Euclidean space,we propose several results of measure theory features for detailed discussion. Utilizing the good structure of set class A_l which consists of those n dimensional left-open and right-closed intervals such that left end point is mi/2l（where miis integer and l is positive integer） and length of each side is 1/2l,combined with partition of real line,inequality techniques and topological techniques, we first prove that Al is a countably infinite partition of n dimensional Euclidean space for each positive integer l,and as l gets larger,Al gets finer,and for each point in each open subset of the n dimensional Euclidean space,the member in Al who contained the point must be contained by the open subset when l is sufficiently large. Then,based on the previous results we prove that every open subset in n dimensional Euclidean space can be expressed as the union of at most countably infinite n dimensional left-open and right-closed intervals by way of contradiction. Last,arming with this theorem,we introduce some generators for the Borel algebra of n dimensional Euclidean Space.