Characterization 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:


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

where:


 * $\sigma \left({f_i: i \in I}\right)$ is the $\sigma$-algebra generated by $\left({f_i}\right)_{i \in I}$
 * $\sigma \left({\displaystyle \bigcup_{i \mathop \in I} f_i^{-1} \left({\Sigma_i}\right)}\right)$ is the $\sigma$-algebra generated by $\displaystyle \bigcup_{i \mathop \in I} f_i^{-1} \left({\Sigma_i}\right)$
 * $f_i^{-1} \left({\Sigma_i}\right)$ denotes the pre-image $\sigma$-algebra on $X$ by $f$

Proof
For each $i \in I$, one has by definition of generated $\sigma$-algebra:


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

which shows that each of the $f_i$ is measurable.

Next, suppose that $\Sigma$ is a $\sigma$-algebra such that each of the $f_i$ is $\Sigma \,/\, \Sigma_i$-measurable.

Then for all $i \in I$, one has:


 * $f_i^{-1} \left({\Sigma_i}\right) \subseteq \Sigma$

and hence by Union is Smallest Superset: Family of Sets:


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

Finally, by Generated Sigma-Algebra Preserves Subset, it follows that:


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

Thus:


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

by definition of the latter.