Measure-preserving transformations

Question

In the following image the last line why does the measure of the set $G$ equal $1$ ?

Answer 1

The text defines $G$ as $G = \bigcap_{n = 1}^{\infty} G_{n}$, where $G_{n} = \bigcup_{k = 0}^{\infty} \xi^{-k}(V_{n})$. It is therein proved that $\mu(G_{n}) = 1$. Recall that if $A$ and $B$ are events/measurable sets and $\mu(A) = \mu(B) = 1$, then $\mu(A \cap B) = 1$. Thus, for each $N \in \mathbb{N}$, $\mu(G_{1} \cap \dots \cap G_{N}) = 1$ (by induction). Therefore, by continuity of measure, \begin{equation*} \mu(G) = \lim_{N \to \infty} \mu(G_{1} \cap \dots \cap G_{N}) = 1. \end{equation*}