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 $\{ M_j \}_{j \in J}$ |