Let $M_j = (X_j, \mathcal{F}_j)$ be a D1108: Measurable space for each $j \in J$.
Let $X : = \prod_{j \in J} X_j$ be the D326: Cartesian product of $\{ X_j \}_{j \in J}$ such that

(i)	$\mathfrak{S} : = \mathfrak{S}(X)$ is the D484: Set of sigma-algebras on $X$
(ii)	$\mathcal{C} : = \mathcal{C}_M(X)$ is the D3801: Set of measurable cylinder sets in $X$ with respect to $M = \{ M_j \}_{j \in J}$

The product sigma-algebra on $X$ with respect to $M = \{ M_j \}_{j \in J}$ is the D11: Set \begin{equation} \sigma \langle \mathcal{C} \rangle = \bigcap \{ \mathcal{F} : \mathcal{C} \subseteq \mathcal{F} \in \mathfrak{S} \} \end{equation}