Showing that the collection of events $\{\omega: X_i(\omega) \in B_i, i = 1,\dots,k\}$ generates $\sigma(X_1,\dots, X_k)$

Question

Let $(X_n)_{n\in\mathbb{N}}$ be a sequence of independent real valued random variables on the probability space $(\Omega, F, P)$. I'm trying to show that the collection $\mathcal{C}$ of events $\{\omega: X_i(\omega) \in B_i, i = 1,\dots,k\}$, for $B_i \in \mathcal{B}(\mathbb{R})$, generates the sigma-algebra $H_k := \sigma(X_1,\dots,X_k)$. I've already shown that $\mathcal{C}$ is a $\pi$-system hence the next step of my proof would be to show that any $X_i, i = 1,\dots,k$, is $\sigma(\mathcal{C})/\mathcal{B}(\mathbb{R})$ measurable. But given that any element $A \in \mathcal{C}$ has the form $A = \{\omega: X_i(\omega) \in B_i, i = 1,\dots,k\} = \bigcap_{i=1}^k\{\omega: X_i(\omega) \in B_i\} = \bigcap_{i=1}^kX_i^{-1}[B_i]$, I don't really know how to proceed. We certainly know that for any fixed $j$, $A = \bigcap_{i=1}^kX_i^{-1}[B_i] \subset X_j^{-1}[B_j]$, but I can't really convince myself of the fact that then $X_j^{-1}[B_j] \in \sigma(\mathcal{C})$. How should I proceed with this proof?

Answer 1

If $B_i=\mathbb R$ for all $i \neq j$ and $B_j=B$ then $\{\omega: X_i(\omega) \in B_i,i=1,2...,k\}=X_j^{-1}(B)$ so $X_j^{-1}(B)\in \mathcal C \subseteq \sigma (\mathcal C)$.