I'm currently reading Theory of Probability and Random Processes by Koralov and Sinai. In Chapter 2, the authors state the definition of a finite-dimensional elementary cylinder and a finite-dimensional cylinder:
Consider a probability space, $(X,\mathcal{G}, P_X)$.
Let $1 \leq n \leq \infty$ and $\Omega$ be the space of sequences of length $n$. A finite-dimensional elementary cylinder is a set of the form $$ A = \{\omega: \omega_{t_1} \in A_1,\ldots, \omega_{t_k} \in A_k\}, $$ where $t_1,\ldots,t_k\geq 1$ and $A_i \in \mathcal{G}$, $1 \leq i \leq k$. A finite-dimensional cylinder is a set of the form $$A = \{\omega: (\omega_{t_1},\ldots,\omega_{t_k}) \in B\},$$ where $t_1,\ldots,t_k \geq 1$ and $B \in \mathcal{G} \times \ldots\times \mathcal{G}$ ($k$ times).
What is the difference between these two definitions?
I don't really understand how these two definitions are different. My understanding is that a finite-dimensional elementary cylinder is essentially the finite Cartesian product of sets in your $\sigma$-algebra. And a finite-dimensional cylinder is an element in the Cartesian product of the entire $\sigma$-algebra, which is just the Cartesian product of some elements in the $\sigma$-algebra.