Let $M_j = (X_j, \mathcal{F}_j)$ be a D1108: Measurable space for each $j \in J$.
An D548: Ordered pair $M = (X, \mathcal{F})$ is a measurable product space with respect to $\{ M_j \}_{j \in J}$ if and only if

(1)	$X = \prod_{j \in J} X_j$ (D326: Cartesian product)
(2)	$\mathcal{F}$ is a D2154: Product sigma-algebra on $X$ with respect to $\{ M_j \}_{j \in J}$