Proving two sigma algebras coincide Let $(S, \mathcal{S})$ be a measurable space and $\Omega= \prod_{i=0}^{\infty} S$, let $X_i:\Omega \to S$ be the "coordinates" and $\theta_n$ the shift operators on $\Omega$: $w=(x_0,x_1,...) \mapsto (x_n,x_{n+1},...)$. Finally let $\mathcal{A}=\sigma(X_n,  n\geq 0)$ and $ \mathcal{G}_n=\sigma(X_n,X_{n+1},...)$. How would you prove that $\mathcal{G}_n=\theta_n^{-1}(\mathcal{A})$? I am puzzled because it seems to me that the antiimage of an element of $\mathcal{A}$ could very well be in $\mathcal{A}$ and not in $\sigma(X_n,X_{n+1},...)$. This claim is in the context of canonical Markov Chains, Ionescu-Tulcea theorem.
 A: It's known (see, e.g. math.stackexchange.com/questions/7881) that
$$f^{-1}(\sigma(C))=\sigma(f^{-1}(C))$$
and that $A$ is generated by cylinder sets $\mathcal{B}$.
Hence $\theta_n^{-1}(\mathcal{A}) = \theta_n^{-1} (\sigma(\mathcal{B})) = \sigma(\theta_n^{-1} (\mathcal{B})) = \sigma( \mathcal{\tilde{B}}) = \mathcal{G}_n $,
where $\tilde{B}$ are cylinder sets, that generate $\mathcal{G}_n$.
Addition:
elements of $\mathcal{B}$ have the next form: $F_{t_1, \ldots, t_N, D_1, \ldots, D_N} = \{(y_0, y_1, \ldots): y_{t_1} \in D_1, y_{t_2} \in D_2, \ldots, y_{t_N} \in D_N \}$, where $D_i$ are Borel sets of real line and $t_i \ge 0$.
Elements of $\tilde{\mathcal{B}}$ have the next form: $\tilde{F}_{s_1, \ldots, s_N, D_1, \ldots, D_N} = \{(y_0, y_1, \ldots): y_{s_1} \in D_1, y_{s_2} \in D_2, \ldots, y_{s_N} \in D_N \}$, where $D_i$ are Borel sets of real line and $s_i \ge n$.
Thus $\theta_n^{-1} (\mathcal{B}) = \mathcal{\tilde{B}}$ because $$\theta_n^{-1} (\mathcal{B}) =  \theta_n^{-1} ( \{ F_{t_1, \ldots, t_n, D_1, \ldots, D_N}\}) = \{ \theta_n^{-1} (F_{t_1, \ldots, t_N, D_1, \ldots, D_N}) \} = \{ \tilde{F}_{t_1+n, \ldots, t_N+n, D_1, \ldots, D_N} \} = \mathcal{\tilde{B}}.$$
Addition2: $$\theta_n^{-1} ( {F}_{t_1, \ldots, t_N, D_1, \ldots, D_N} ) = \{ (y_0, y_1, \ldots) : \theta_n (y_0, y_1, \ldots) \in {F}_{t_1, \ldots, t_N, D_1, \ldots, D_N}  \}$$
$$= \{ (y_0, y_1, \ldots) :  (y_n, y_{n+1}, \ldots)  \in {F}_{t_1, \ldots, t_N, D_1, \ldots, D_N}  \} = $$
$$ = \{ (y_0, y_1, \ldots) :  y_{n+t_1} \in D_1, y_{n+t_2} \in D_2, \ldots  \}  = \tilde{F}_{t_1+n, \ldots, t_N+n, D_1, \ldots, D_N}.$$
