# Measure-preserving transformation of random sequence

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

$$$\mathbb{P}\left[\omega \in \Omega\vert \left(X_t(\omega)\right)_{t \in \mathbb{N}} \in E\right] = \mathbb{P}\left[\omega \in \Omega\vert \left(X_t(\mathsf{T}\omega)\right)_{t \in \mathbb{N}} \in E\right],\; \forall E\in \mathscr{B}^{\infty}$$$

or is it

$$$\mathbb{P}\left[\omega \in \Omega\vert \left(X_t(\omega)\right)_{t \in \mathbb{N}} \in E\right] = \mathbb{P}\left[\mathsf{T}\omega \in \Omega\vert \left(X_t(\mathsf{T}\omega)\right)_{t \in \mathbb{N}} \in E\right],\; \forall E\in \mathscr{B}^{\infty}$$$

I am failing to make the connection here.

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The second formula is a tautology since $\{\omega\in\Omega\mid Y(\omega)\in E\}=\{T\omega\in\Omega\mid Y(T\omega)\in E\}$ for every $Y$ defined on $\Omega$ and every onto $T:\Omega\to\Omega$. To wit, the second set can be rewritten as $$\{\alpha\in\Omega\mid\exists\omega\in\Omega,\,\alpha=T\omega,\, Y(\alpha)\in E\}.$$ If the condition that there exists $\omega\in\Omega$ such that $\alpha=T\omega$ is fulfilled for every $\alpha\in\Omega$, to cancel it does not change the definition.
The first definition (the correct one) says that the distributions of $Y$ and $Y\circ T$ coincide, where $T:(\Omega,\mathscr F)\to(\Omega,\mathscr F)$ should be measurable and where in your case it happens that $Y=(X_t)_t$ hence $Y$ is a random variable $Y:(\Omega,\mathscr F,\mathbb P)\to(\mathbb R^{\infty},\mathscr B^{\infty})$, but in fact, $Y$ could be any random variable $Y:(\Omega,\mathscr F,\mathbb P)\to(E,\mathscr E)$ for any measurable space $(E,\mathscr E)$.
This is also equivalent to the fact that the distribution of $Y$ is the same with respect to $\mathbb P$ and $\mathbb Q$, where $\mathbb Q$ is the probability measure defined on $\mathscr F$ by $\mathbb Q(A)=\mathbb P(T^{-1}(A))$ (a definition which makes sense as soon as $T:(\Omega,\mathscr F)\to(\Omega,\mathscr F)$ is measurable).