Ergodic transformations that generate the Carathéodory definition of measurable sets

Motivated by a pedagogical argument about the existence of intuitive interpretations of the important definition of measurable sets given by Caratheodory [7] 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.