(Durrett, Probability : Theory and Examples) Exercise 7.1.4

Question

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 $\mathcal{G}$

Answer 1

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.