Let $\{X_t\}, t\in [0,\infty)$ be a collection of random variables and $C_1 = \{ E \mid E = \{ (X_{t_1}, X_{t_2}, X_{t_3}, \cdots, X_{t_n}) \in B\},n \in \mathbb{N},B\in \mathcal{B}(\mathbb{R}^n)\}$ where $t_i \in [0,\infty)~\forall i$.
$C_1$ can be shown to be an algebra (field) basically by "embedding" the Borel sets in higher dimensions.
Now, if the definition is changed slightly to:
$C_2 = \{ E \mid E = \{ (X_{t_1}, X_{t_2}, X_{t_3}, \cdots, X_{t_n}) \in B\},\\n \in \mathbb{N},B\in\mathcal{B}(\mathbb{R}^n), 0 \leq t_1 < t_2 < t_3 \cdots < t_{n-1} < t_n\}\}.$
Is $C_2$ still an algebra? The issue is that the Borel sets some how have to be interspersed and I am not sure how to do this.
Thanks, Phanindra