Nonmeasurable Set and the Cardinality of Its Class
Xiao-Min Hu · 2005
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 cardinality of the class of open sets.After studying the construction of nonmeasurable set,the paper gives out an example of nonmeasurable set,by assuming the Zermelo 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 c_0 and c holds.Finally,the paper proves that the class of nonmeasurable sets has the cardinality of c.