| 摘要: |
| 针对n维欧氏空间上Borel〖WTBX〗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/2l（其中mi为整数，l为正整数），且长度均为1/2l的那些左开右闭区间形成的集类Al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Al是n维欧氏空间的可数无限划分，且随着l变得越大Al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，Al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〖WTBZ〗Borel代数的几个较小生成元. |
| 关键词: n维欧氏空间 Borel代数 较小生成元 |
| DOI： |
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| A Brief Analysis on the Construction of Borel Sets in n Dimensional Euclidean Space |
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ZENG Xiao lin1，HUANG Yi yuan2
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| Abstract: |
| 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 Al which consists of those n dimensional left open and right closed intervals such that left end point is mi/2l（where mi is 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. |
| Key words: n dimensional Euclidean space Borel algebra generator |
