Product Sigma Algebra
# Definition
Let $(X_{\alpha}, \mathcal{M}\alpha){\alpha \in I}$ be a family of Measure Spaces indexed by Index Set $I$. The Product Sigma Algebra on $\prod\limits_{\alpha \in I} X_{\alpha}$ is defined as $$\bigotimes\limits_{\alpha \in I} \mathcal{M}{\alpha} = \sigma ({\pi{\alpha}^{-1}(E_\alpha): \forall \alpha \in I, \forall E_{\alpha}\in \mathcal{M}\alpha})$$ where $\pi{\alpha} : \prod\limits_{\beta \in I} X_{\beta} \to X_\alpha$ is the $\alpha$-Projection Map.