Connected sets of Van Vleck

Abstract

Both Vitali and Van Vleck have given interesting constructions of Lebesgue nonmeasurable sets in euclidean E1. Here we give a generalization for the construction of Van Vleck for Em, m > 2; our interest is in the type of connected set that can be so obtained. Elsewhere we will consider the construction of Vitali. Of interest also is the interlacing of these connected sets. Below Q is the first transfinite ordinal whose cardinal is the same as that of the linear continuum: a, ,B, y are ordinals, >0 and <Q. We will say that the Van Vleck basic set for a given poitnt Pa =(xia, X2a, X * Xma) in (x1, X2, * * *, xm)-coordinate space is the set of all (x',, x2a, x, x,,) where for each j (j= 1, 2, m, i), we have as in [1, p. 240],