Definition:Sigma-Algebra Generated by Collection of Mappings

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Definition

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 the $\sigma$-algebra generated by $\left({f_i}\right)_{i \in I}$, $\sigma \left({f_i: i \in I}\right)$, is the smallest $\sigma$-algebra on $X$ such that every $f_i$ is $\sigma \left({f_i: i \in I}\right) \, / \, \Sigma_i$-measurable.

That is, $\sigma \left({f_i: i \in I}\right)$ is subject to:

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


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


Also see


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