what does $(\Omega^T,\mathcal{A}^T)$ mean? Let $(\Omega_t,\mathcal{A}_t), t\in T$ be a collection of measurable spaces. What does the notation mean? $(\Omega^T,\mathcal{A}^T)$
 A: The set $\Omega^T\triangleq \{ (\omega_t)_{t\in T} : \omega_t\in\Omega_t\}$ is the set of all configurations $\omega\triangleq  (\omega_t)_{t\in T}$.  


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*If $T$ is finite say $T = \{1, \ldots, n\}$ then $\Omega^T=\Omega^{\{1, \ldots, n\}}=\Omega_1\times\ldots\times\Omega_n$ is the set of all the n-tuples $(\omega_t)_{t\in \{1,\ldots,n\}}=(\omega_1,\ldots\omega_n)$.

*If $T$ is the set $\mathbb{N}$ of natural numbers then $\Omega^T=\Omega^{\mathbb{N}}=\Omega_1\times\ldots\times\Omega_n\times\ldots\quad $ is the set of all infinite sequences $(\omega_n)_{n\in \mathbb{N}}=(\omega_1,\ldots\omega_n,\ldots \;)$.

*If T is the set $\mathbb{R}$ of real numbers then $\Omega^T=\Omega^{\mathbb{R}}$ is the set of all applications $T=\mathbb{R}\ni t\mapsto \omega_t\in\bigcup_{t\in\mathbb{R}}\Omega_t$.

*Generally, if $T$ is any arbitrary set then the configurations $(\omega_t)_{t\in T}$ are nothing more than functions of $T$ to $\bigcup_{t\in T}\Omega_t$.
Most times the sigma field $\mathcal{A}^T$ is the sigma field of cylinders. You can find a beautiful exposition of the sigma of the cylinders in the book of Robert Ash.
