  It is known that there are specific examples of ergodic transformations on
measure spaces for which the calculation of the outer measure of transformation
invariant sets leads to a condition closely resembling Carath\'eodory's
condition for sets to be measurable. It is then natural to ask what functions
are capable of `generating', that is leading to, the Carath\'eodory definition
in the same way. The present work answers this question by showing that the
property of generating Carath\'eodory's definition holds for the general class
of outer measure preserving ergodic transformations on measure spaces. We
further show that the previously found specific examples of functions
generating Carath\'eodory's definition fall into this family of
transformations.
