Note: This question is related to this question about the construction of stochastic processes. Specifically, it relates to the transformation $\mathbb S: \Omega \rightarrow \Omega$ that is mentioned. The following is an example to help understand such transformations. If such transformation are measure preserving, then the distribution function of $X_t$ is identical for all $t \geq 0$.
If we suppose that $\Omega = [0,1)$ and that $Pr$ is the uniform measure and that $$ \mathbb S(\omega) = \begin{cases} 2 \omega & \omega \in [0, 1/2) \\ 2 \omega - 1 & \omega \in [1/2, 1), \end{cases} $$ how would we show that $\mathbb S$ is measure-preserving?
It definitely suffices to verify this property for just the open intervals in $\Omega$. But trying to so explicitly for an arbitrary interval $(a,b) \subset \Omega$ is complicated by the fact that it's hard to say more than $\mathbb S((a,b)) \subset [0,1)$ when $(a,b) \subset [0,1/2)$ and that again $\mathbb S((a,b)) \subset [0,1)$ when $(a,b) \subset [1/2)$. Given this, what is an easy way to go about arguing that $\mathbb S$ is measure-preserving?
