A basic question really. Consider the infinite sequence of random variable $\left(X_t\right)_{t \in \mathbb{N}}$ where $X_t:\Omega \mapsto \mathbb{R},\, \forall t$ and the spaces are $\left(\Omega, \mathscr{F}, \mathbb{P}\right)$ and $\left(\mathbb{R}, \mathscr{B}, \mathbb{P}_X\right)$. The random sequence induces the product topology $\left(\mathbb{R}^{\infty}, \mathscr{B}^{\infty}\right)$ generated by the finite dimensional cylinders. Also, we have one-one and onto transformation $\mathsf{T}:\Omega \mapsto \Omega$.
I want to know what the appropriate definition of a measure-preserving transformation is. Is it
$\begin{equation} \mathbb{P}\left[\omega \in \Omega\vert \left(X_t(\omega)\right)_{t \in \mathbb{N}} \in E\right] = \mathbb{P}\left[\omega \in \Omega\vert \left(X_t(\mathsf{T}\omega)\right)_{t \in \mathbb{N}} \in E\right],\; \forall E\in \mathscr{B}^{\infty} \end{equation}$
or is it
$\begin{equation} \mathbb{P}\left[\omega \in \Omega\vert \left(X_t(\omega)\right)_{t \in \mathbb{N}} \in E\right] = \mathbb{P}\left[\mathsf{T}\omega \in \Omega\vert \left(X_t(\mathsf{T}\omega)\right)_{t \in \mathbb{N}} \in E\right],\; \forall E\in \mathscr{B}^{\infty} \end{equation}$
I am failing to make the connection here.