Master of Science
School of Mathematics and Applied Statistics
Koeller, Amos Nathan, Ergodic transformations that generate the Carathéodory definition of measurable sets, Master of Science thesis, School of Mathematics and Applied Statistics, University of Wollongong, 2003. https://ro.uow.edu.au/theses/2714
Motivated by a pedagogical argument about the existence of intuitive interpretations of the important definition of measurable sets given by Caratheodory  in 1914 this thesis defines the concept of a transformation which generates the definition of Caratheodory, and describes transformations which posses this property. In doing so, particular attention is paid to the relatively uninvestigated behaviour of the outer measures of sets invariant under ergodic transformations. The investigation of sets invariant under ergodic transformations also considers the ergodic tool of tower extensions of transformations and their effect on transformation invariant sets, the transformation properties of invariance, mixing (in the ergodic sense) and the generation of Caratheodory's definition. Specific examples of transformations coming within the orbit of the discussion are irrational rotations on the unit circle, Kakutani's transformation, Chacon's tower transformation and generalisations of some of these.