In my lecture notes there is the following:
The $\sigma-$algebra that is produced from a family of random variables $\{\xi_i, i \in I\}$ (where the set of indices $I$ can be finite or infinite) is defined as the smallest $\sigma-$algebra that contains the sets of the form $\xi_i^{-1}(B)$ where $B \in \mathcal{B}(\mathbb{R})$ and $i \in I$.
$$$$
First question:
Isn't each $\sigma-$algebra procuded by a family of random variables???
Second question:
Isn't there also an other $\sigma-$algebra that contains elements of the form $\xi_i^{-1}(B)$ other than the smallest one???