On the class of perfectly null sets and its transitive version

TL;DR

This paper introduces new classes of null sets on the real line, explores their properties, and investigates their relationships with existing null set classes, highlighting open questions and consistency results.

Contribution

It defines the classes of perfectly null sets and transitive perfectly null sets, analyzing their properties and their relation to strongly null and universally null sets.

Findings

Every strongly null set is transitive perfectly null.

It is consistent with ZFC that a universally null set is not transitive perfectly null.

Open questions remain about the equivalence of these classes.

Abstract

We introduce two new classes of special subsets of the real line: the class of perfectly null sets and the class of sets which are perfectly null in the transitive sense. These classes may play the role of duals to the corresponding classes on the category side. We investigate their properties and, in particular, we prove that every strongly null set is perfectly null in the transitive sense, and that it is consistent with ZFC that there exists a universally null set which is not perfectly null in the transitive sense. Finally, we state some open questions concerning the above classes. Although the main problem of whether the classes of perfectly null sets and universally null sets are consistently different remains open, we prove some results related to this question.