Let $(\Omega,\mathbb{F},P)$ be a probability space and $\epsilon_1,...,\epsilon_n$ be real-valued random variables defined on $\Omega$. Now let $\mathbb{D}_n$ be the sigma-algebra generated by $\epsilon_1,...,\epsilon_n$, i.e $ \mathbb{D_n}=\sigma(\epsilon_1,...,\epsilon_n)= \sigma \left( \bigcup_{i=1}^n \{ \epsilon_i^{-1}(B)|B\in \mathbb{B} \} \right) $ where $\mathbb{B}=\mathbb{B(\mathbb{R})}$ is the Borel sigma-algebra. With $X_0=0$ define recursively $ X_k=\alpha X_{k-1}+\epsilon_k \quad \quad \quad \text{for} \quad k=1,...,n$ I have shown that each $X$ can be defined as $ X_k=\sum_{i=0}^{k-1} \alpha^i\epsilon_{k-i}=\epsilon_k+\alpha \epsilon_{k-1}+ \cdot \cdot \cdot + \alpha^{k-1}\epsilon_1 $ Now the problem is that i am to show that $ \sigma(\epsilon_1,...,\epsilon_k)= \sigma(X_1,...,X_k) \quad \quad \quad \text{for} \quad k=1,...,n$ So far my strategy for showing the above is to first show that $\left(\bigcup_{i=1}^k \{\epsilon_i^{-1}(B)|B\in\mathbb{B}\} \right) \subseteq \left(\bigcup_{i=1}^k \{X_i^{-1}(B)|B\in\mathbb{B}\} \right) $ and conversely $\supseteq$, by taking an arbitrary element in the first and showing that it is in the other.
Now the great question: How is this done?