Definition 23.2.1 (Product $\sigma$-Algebra).label Let $\bracs{(X_i, \cm_i)}_{i \in I}$ be measurable spaces, then the product $\sigma$-algebra $\bigotimes_{i \in I}\cm_{i}$ on $\prod_{i \in I}X_{i}$ is the $\sigma$-algebra generated by $\seqi{\pi_i}$.