 Measure preserving versus non-measure preserving transformations. Roughly speaking, a transformation is measure preserving if the size of a set (like set \(A\) in the above example) does not change over the course of time: a set can change its form but it cannot shrink or grow (with respect to the measure). Formally, \(T\) is a measure-preserving transformation on \(X\) if and only if (iff) for all sets \(A\) in \(X: \mu(A) = \mu(T^{-1}(A))\), where \(T^{-1}(A)\) is the set of points that gets mapped onto \(A\) under \(T\); that is \(T^{-1}(A) = \{ x \in X \mid T(x) \in A  \}\).[4]  From now on we also assume that the transformations we consider are measure  preserving.[5]