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

Theorem

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 $\map \sigma {f_i: i \in I}$, the $\sigma$-algebra generated by $\family {f_i}_{i \mathop \in I}$, exists and is unique.

Proof

By Characterization of Sigma-Algebra Generated by Collection of Mappings:

$\ds \map \sigma {f_i: i \in I} = \map \sigma {\bigcup_{i \mathop \in I} \map {f_i^{-1} } {\Sigma_i} }$

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.

$\blacksquare$

Sources

• 2005: René L. Schilling: Measures, Integrals and Martingales ... (previous) ... (next): $7.5$