Classifying sets of measure zero with respect to their open covers

Abstract:

Developing ideas of Borel and Fréchet, we define a partial preorder which classifies measure zero sets of reals according to their open covers and study the induced partial order on the equivalence classes. The more "rarefied" a set of measure zero, the higher it will range in our partial order. Main results: The sets of strong measure zero form one equivalence class that is the maximum element of our order. There is a second highest class that contains all uncountable closed sets of measure zero. There is a minimum class that contains all dense ${G_\delta }$-subsets of the real line of measure zero. There exist at least four classes, and if Martin’s axiom holds, then there are as many classes as subsets of the real line. It is also consistent with ZFC that there is a second lowest class.