The problem is presented in Example 1.63 (page 29) of "Probability Theorey" 3rd edition by Prof. Achim Klenke.
We construct a measure for an infinitely often repeated random experiment with finitely many possible outcomes.
Notations:
Let $E$ be the set of possible outcomes. For a fixed realization of the repeated experiment, let $\omega_{1},\omega_{2},\ldots \in E$ be the observed outcomes. Hence the space of all possible outcomes of the repeated experiment is $\Omega = E^{\mathbb{N}}$.
Define the set of all sequences whose first $n$ values are $\omega_{1},\ldots,\omega_{n}$: \begin{equation*} [\omega_1, \ldots, \omega_n]:=\{\omega^{\prime} \in \Omega: \omega_i^{\prime}=\omega_i \text { for any } i=1, \ldots, n\} \end{equation*} Let $\mathcal{A}_0=\{\emptyset\}$. For $n \in N$, define the class of cylinder sets that depend only on the first $n$ coordinates \begin{equation*} \mathcal{A}_n:=\{[\omega_1, \ldots, \omega_n]: \omega_1, \ldots, \omega_n \in E\}, \end{equation*} and let $\mathcal{A} :=\bigcup_{n=0}^{\infty} \mathcal{A}_n$.
Purpose:
Let $A, A_1, A_2, \ldots \in \mathcal{A}$ and $A \subset \bigcup_{n=1}^{\infty} A_n$. We want to show that there exists an $N \in \mathbb{N}$ such that \begin{equation*} A \subset \bigcup_{n=1}^N A_n . \end{equation*}
Proof
Let $B_n:=A \backslash \bigcup_{i=1}^n A_i$. We assume $B_n \neq \emptyset$ for all $n \in N$ in order to get a contradiction. By Dirichlet's pigeonhole principle (recall that $E$ is finite), we can choose $\omega_1 \in E$ such that $\left[\omega_1\right] \cap B_n \neq \emptyset$ for infinitely many $n \in N$. Since $B_1 \supset B_2 \supset \cdots$, we obtain \begin{equation*} [\omega_1] \cap B_n \neq \emptyset \quad \text { for all } n \in N . \end{equation*} Successively choose $\omega_2, \omega_3, \ldots \in E$ in such a way that \begin{equation*} [\omega_1, \ldots, \omega_k] \cap B_n \neq \emptyset \quad \text { for all } k, n \in N \end{equation*}
Question:
Can we now directly conclude that \begin{equation*} \omega= (\omega_1, \omega_2, \ldots) \in B_{n} \text{ for all } n \in \mathbb{N} \end{equation*} Is there any danger of it?