Automorphism groups of measures on the Cantor space. Part I: Good measures and Rokhlin properties

We study criteria for the existence of a dense or comeager conjugacy class in the automorphism group of a given measure on the Cantor space. We concentrate on good measures, defined by Akin [\emph{Trans.\ Amer.\ Math.\ Soc.} $\mathbf{\text{357}}$ (2005), no. 7, 2681–2722], which we characterize as a particular subclass of ultrahomogeneous measures. We determine good measures with rational values on clopen sets whose automorphism group admits a comeager conjugacy class. Our approach uses the Fraïssé theory.