关于Borel集的一个注记

A Note on the Borel Set

从教科书中关于Borel集定义的两种不同叙述法入手,应用超限归纳法,证明了这两种定义的等价性,并根据基数理论,证明了非Borel的Lebesgue可测集的存在性,进而证明了R1上非Borel的Lebesgue可测集的基数是2c-c,其中c为连续统基数。

Two different statements of Borel set of R1 were referred in textbooks. Transfinite induction was used in this paper to prove the equivalence of the two different statements. Based on Z-F Axiom Systems of set theory,the paper proved the existence of the measurable set of non-Borel of Lebesgue. Furthermore, it showed that the measurable set of non-Borel of Lebesgue on R1 was 2c- c, and c is the cardinal number of continuum.