# (Durrett, Probability : Theory and Examples) Exercise 7.1.4

7.1.4 $$A \in F_o$$ if and only if there is a sequence of times $$t_1,t_2,... \in[0,\infty)$$ and a $$B \in \mathcal{R}^{\{1,2,...\}}$$ so that $$A={\omega: (\omega(t_1),\omega(t_2),...)\in B}$$. In words, all events in $$F_o$$ depend on only countably many coordinates

$$\mathcal{F}_o$$ is $$\sigma$$-field generated by the finite dimentsional sets $$\{w:w(t_i)\in A_i \text{ for } 1\leq i\leq n\}$$ where $$A_i \in \mathcal{R}$$

To prove the only if part, the solutions says that it is is enough to show that $$\mathcal{G}$$ is $$\sigma$$-field where $$\mathcal{G}$$ is defined as below $$\mathcal{G}=\{A=\{w:(w(t_1),w(t_2),...)\in B\}:B\in\mathcal{R}^{1,2,...}\}$$

I can't understand why proving that $$\mathcal{G}$$ is $$\sigma$$-field is sufficient and the relation of $$\mathcal{F}_o$$ and $$\mathcal{G}$$

I think what's intended is that $$\mathcal G$$ be the collection of all sets that satisfy the second part (i.e. the collection of all sets $$A$$ such that there is a sequence of times $$t_1,t_2,\ldots$$ and a $$B\in \mathcal R^{1,2,\ldots}$$ so that $$A = \{w:(w(t_1),w(t_2),\ldots)\in B\}$$). In other words, the problem is to show that $$F_o=\mathcal G,$$ and the "only if" part of the problem is to show $$F_o\subseteq \mathcal G.$$ This can be accomplished by showing that $$\mathcal G$$ is a $$\sigma$$-field containing all the finite-dimensional sets, since $$F_o$$ is the $$\sigma$$-field generated by those sets. $$\mathcal G$$ contains all the finite-dimensional sets, since $$A_1\times \ldots \times A_n \times \mathbb R\times\mathbb R\times\ldots\in \mathcal R^{1,2\ldots},$$ so what's left is to show $$\mathcal G$$ is a $$\sigma$$-field.
• How can we conclude $F_o \subset \mathcal{G}$ if $\mathcal{G}$ is $\sigma$-field containing all the finite dimensional sets and $F_o$ is the $\sigma$-field generated by the finite dimentional sets? May 5, 2021 at 6:08
• @Dongri That's what it means to be generated by something. It is the smallest $\sigma$-field containing that something, i.e. the intersection of every $\sigma$-field containing that something. May 5, 2021 at 6:16