I am studying the book "Probability Theory" from Achim Klenke.
In this book the Measure extension Theoremis constructed in the following way:
Consider an infinitely often repeated random experiment with finitely many possible outcomes. Let $E$ be the set of possible outcomes. Let $\omega_{1},\omega_{2},....\in E$ be the observed outcomes. Hence the space of all possible outcomes of the repeated experiment is $\Omega=E^\mathbb{N}$. Then they defined the set of all sequences whose first n values are $\omega_{1},....\omega_{n}$ as $\mathcal{A}_{n}:= \{[\omega_{1},....\omega_{n}]:\omega_{1},....\omega_{n} \in E\}$ Then $\mathcal{A}:=\bigcup_{n=0}^{\infty}\mathcal{A}_{n}$ forms a semi-ring.
Now they defined a probability measure on the semi-ring $\mathcal{A}$ and want extend it to $\sigma(\mathcal{A})$. For that purpose they want to prove first the existence of $\sigma(\mathcal{A})$ in the following way:
Define the (ultra-)metric $d$ on $\Omega$ by
$d(\omega,\omega^{'})= \left\{\begin{array}{@{}lr@{}} 2^{-inf\{n\in\mathbb{N}:\omega_{n}\ne\omega_{n}^{'}\}} & \text{if }\omega\ne\omega^{'} \\ 0 & \text{if}\omega=\omega^{'} \end{array}\right\}$
They claim $(\Omega,d)$ is a compact metric space and $[\omega_{1},....\omega_{n}]=B_{2^{-n}}(\omega)=\{\omega^{'}\in\Omega:d(\omega, \omega_{'}) <2^{-n}\}$.
The complement of $[\omega_{1},....\omega_{n}]$ is an open set, as it is the union of $(\#E)^{n}-1$ open balls $[\omega_{1},....\omega_{n}]^{c}=\bigcup_{(\omega_{1}^{'},....\omega_{n}^{'})\ne(\omega_{1},....\omega_{n})}[\omega_{1}^{'},....\omega_{n}^{'}]$. Since $\Omega$ is compact, the closed subset $[\omega_{1},....\omega_{n}]$ is compact. Therefore $\sigma(\mathcal{A})=\mathcal{B}(\Omega,d)$, which guaranteed the existence. Applying Caratheodory Theorem, we obtain an extension on the Sigma-Algebra.
Now the following Point I couldn't follow:
- $[\omega_{1},....\omega_{n}]$ is defined as an open-ball, why it is closed?
- Why is $(\Omega,d)$ compact?
- I cannot see why $\sigma(\mathcal{A})=\mathcal{B}(\Omega,d)$. The Borel-Sigma-Algebra is defined as the sigma-algebra generated by Open-sets. I know it is equivalent to the sigma-algebra generated by closed sets and compact sets. However, the set $\mathcal{A}$, as a countable union of $[\omega_{1},....\omega_{n}]$, which is closed and compact, can still not be classified as open, close or compact.