What is $\sigma\{X_j ; j=0,...n\}$ where $X_j$ are independent random variables

Question

What is $F_n = \sigma\{X_j ; j=0,...n\}$ where $X_j$ are independent $L^1(\Omega,F,P)$.

Am I correct in saying that an event $A$ in the sigma algebra $F_n$ is like: $\{w\epsilon\Omega; X_j\epsilon A_j\}$. {So every A in $F_n$ is a unique choice of $X_j$ takes on for $j\epsilon\{0,...n\}$

Answer 1

$$\sigma(X_0,X_1,...,X_n)=\{(X_0,X_1,...,X_n)^{-1}(A): A \text { is Borel in } \mathbb R^{n+1}\}$$ $$=\{\omega: (X_0(\omega),X_1(\omega),...,X_n(\omega)) \in A: A \text { is Borel in } \mathbb R^{n+1}\}.$$