Existence and Uniqueness of Sigma-Algebra Generated by Collection of Mappings

Theorem
Let $\left({X_i, \Sigma_i}\right)$ be measurable spaces, with $i \in I$ for some index set $I$.

Let $X$ be a set, and let, for $i \in I$, $f_i: X \to X_i$ be a mapping.

Then $\sigma \left({f_i: i \in I}\right)$, the $\sigma$-algebra generated by $\left({f_i}\right)_{i \in I}$, exists and is unique.

Proof
By Characterization of Sigma-Algebra Generated by Collection of Mappings, we have that:


 * $\sigma \left({f_i: i \in I}\right) = \sigma \left({\displaystyle \bigcup_{i \mathop \in I} f_i^{-1} \left({\Sigma_i}\right)}\right)$

where the second is a $\sigma$-algebra generated by a collection of subsets.

The result follows from applying Existence and Uniqueness of Sigma-Algebra Generated by Collection of Subsets.