Some possible covers of measure zero sets

Abstract

n ⋃ k≥n Ik. A partial ordering was introduced in [8] to identify the measure zero sets with similar covering properties, and it was shown that at least four essential differences exist. In [10], it was shown that this number cannot be improved using the usual axioms of set theory (ZFC); more precisely, we showed that there are exactly four classes of measure zero sets assuming u < g, a forcing axiom known to be relatively consistent with ZFC (see [2]); we describe here explicitly (what should have been done in [10]) the covering properties of those four classes. In [4], Borel defined regular measure zero sets in order to extended Weierstrass’ theory of analytic functions. They are equipped with special covers and Borel needed a regular measure zero set with a fast enough rate of convergence for the series of lengths of one of its covers to pursue his theory. In an attempt to understand which measure zero sets could satisfy Borel’s condition, we investigate the rate of convergence of these covers compared with the previous classification; under u < g, both types of covers have the same properties.