Characterization of Sigma-Algebra Generated by Collection of Mappings
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Theorem
Let $\struct {X_i, \Sigma_i}$ 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:
- $\map \sigma {f_i: i \in I} = \map \sigma {\ds \bigcup_{i \mathop \in I} \map {f_i^{-1} } {\Sigma_i} }$
where:
- $\map \sigma {f_i: i \in I}$ is the $\sigma$-algebra generated by $\family {f_i}_{i \mathop \in I}$
- $\map \sigma {\ds \bigcup_{i \mathop \in I} \map {f_i^{-1} } {\Sigma_i} }$ is the $\sigma$-algebra generated by $\ds \bigcup_{i \mathop \in I} \map {f_i^{-1} } {\Sigma_i}$
- $\map {f_i^{-1} } {\Sigma_i}$ denotes the pre-image $\sigma$-algebra on $X$ by $f$
Proof
For each $i \in I$, one has by definition of generated $\sigma$-algebra:
- $\ds \map {f_i^{-1} } {\Sigma_i} \subseteq \bigcup_{i \mathop \in I} \map {f_i^{-1} } {\Sigma_i} \subseteq \map \sigma {\bigcup_{i \mathop \in I} \map {f_i^{-1} } {\Sigma_i} }$
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:
- $\map {f_i^{-1} } {\Sigma_i} \subseteq \Sigma$
and hence by Union is Smallest Superset: Family of Sets:
- $\ds \bigcup_{i \mathop \in I} \map {f_i^{-1} } {\Sigma_i} \subseteq \Sigma$
Finally, by Generated Sigma-Algebra Preserves Subset, it follows that:
- $\ds \map \sigma {\bigcup_{i \mathop \in I} \map {f_i^{-1} } {\Sigma_i} } \subseteq \Sigma$
Thus:
- $\ds \map \sigma {\bigcup_{i \mathop \in I} \map {f_i^{-1} } {\Sigma_i} } = \map \sigma {f_i : i \in I}$
by definition of the latter.
$\blacksquare$
Sources
- 2005: René L. Schilling: Measures, Integrals and Martingales ... (previous) ... (next): $7.5$
- 2005: René L. Schilling: Measures, Integrals and Martingales ... (previous) ... (next): $\S 7$: Problem $3 \ \text{(ii)}$