Definition:Sigma-Algebra Generated by Collection of Mappings

Definition

Let $I$ be an indexing set.

Let $\family {\struct {X_i, \Sigma_i} }_{i \mathop \in I}$ be a family of measurable spaces.

Let $X$ be a set.

Let $\family {f_i: X \to X_i}_{i \mathop \in I}$ be a family of mappings.

Then the $\sigma$-algebra generated by $\family {f_i}_{i \mathop \in I}$, $\map \sigma {f_i: i \in I}$, is the smallest $\sigma$-algebra on $X$ such that every $f_i$ is $\map \sigma {f_i: i \in I} \, / \, \Sigma_i$-measurable.

That is, $\map \sigma {f_i: i \in I}$ is subject to:

$(1):\quad \forall i \in I: \forall E_i \in \Sigma_i: \map {f_i^{-1} } {E_i} \in \map \sigma {f_i: i \in I}$
$(2):\quad \map \sigma {f_i: i \in I} \subseteq \Sigma$ for all $\sigma$-algebras $\Sigma$ on $X$ satisfying $(1)$

In fact, $\map \sigma {f_i: i \in I}$ always exists, and is unique, as proved on Existence and Uniqueness of Sigma-Algebra Generated by Collection of Mappings.