一维不可测集及其类的势

Nonmeasurable Set and the Cardinality of Its Class

该文主要探讨一维不可测集的构造以及一维不可测集全体所组成的类的势。为了得到主要结果,讨论了一维开集全体组成的类的势,证明了一维开集全体组成的类具有连续势;讨论了一维不可测集的构造,在承认策莫罗选择公理的前提下,依据Lebesgue测度的平移不变性,用构造的方法给出了一维不可测集,从而说明了一维不可测集的存在性;并用类似的构造方法证明了任何正测度集都具有不可测子集;在承认Cantor连续统假设的前提下,说明了任何一维不可测集都具有连续势;最后,证明了一维不可测集全体所组成的类的势为2c。

This paper mainly concerns the construction of one - dimension nonmeasurable set and the cardinality of its class. For this purpose the paper proves that the class of all open sets has the cardinality of c, which follows studying the eardinality of the class of open sets. After studying the constrnction of nonmeasurable set, the paper gives out an example of nonmeasurable set, by assuming the Zennelo Axiom of choice holds, basing on the invariance of Lebesgue measure under translation modulo 1, using the constructive method which is used to prove every positive set has an nonmeasurable subset, latter. The paper also proves that every, nonmeasurable set has the cardinality of c, by assuming that Cantor＇s hypothesis of there being no other cardinality between co and c holds. Finally, the paper proves that the class of nonmeasurable sets has the eardinality of c.