Definition 15.1.5 (Generated $\sigma$-Algebra). Let $X$ be a set, $\bracs{(Y_i, \cn_i)}_{i \in I}$ be measurable spaces, and $\seqi{f}$ with $f_{i}: X \to Y_{i}$ for each $i \in I$. The $\sigma$-algebra generated by $\seqi{f}$,
\[\sigma(\bracs{f_i| i \in I}) = \sigma\paren{f_i^{-1}(\cn_i)|i \in I}\]
is the smallest $\sigma$-algebra on $X$ such that each $\seqi{f}$ is measurable.