广义Vitali不可测集的构造

Constructions of Generalized Vitali's Nonmeasurable Sets

将构造实数直线R上的Lebesgue不可测集的Vitali方法推广到R的可数子环,证明了R的不与Z同构的可数子环均在R中稠密,进而证明了相应于可数子环H的广义Vitali集是不可测集当且仅当H不含于整数环Z;证明了广义Vitali不可测集的内测度均是0,而外测度可以是任意正数.

This short note is devoted to a new construction of the generalized Vitali＇s nonmeasurable sets on the real line R, that is, replacing the rationals to arbitray infinitely enumerahle suhring of R. It is proved that any nontrivial subring of R not isomophic to the integer ring Z is dense in R, whence the corresponding choice of the generalized Vitali-set of an enumerahle suhring H is nonmeasurable if and only if H is not contained in Z. It is further proved that the inner measure of any generalized Vitali set is zero while its outer measure can be chosen to be any positive number.