Haar Null Sets in the Space of Automorphisms on [0,1]

Abstract

In this paper we study the distinctness of the concepts of left transverse, right transverse and transverse in the space ${\cal H}[0, 1]$ of automorphisms on $[0, 1]$. We show that, in ${\cal H}[0, 1]$, the existence of a Borel probability measure left transverse and right transverse to a set $Y$ does not imply the set $Y$ is Haar null and we exhibit an uncountable disjoint collection of Borel sets that are not Haar null.