Let $M_j = (X_j, \mathcal{F}_j)$ be a D1108: Measurable space for each $j \in J$.
Let $X = \prod_{j \in J} X_j$ and $\mathcal{F} = \prod_{j \in J} \mathcal{F}_j$ each be a D326: Cartesian product.
Let $\mathcal{P}_{\mathsf{cofinite}}(J)$ be the D2200: Set of cofinite sets in $J$.
A D11: Set $\prod_{j \in J} E_j \subseteq \mathcal{F}$ is a measurable cylinder set in $X$ with respect to $M = \{ M_j \}_{j \in J}$ if and only if
\begin{equation}
\exists \, I \in \mathcal{P}_{\mathsf{cofinite}}(J) : \forall \, i \in I : E_i = X_i
\end{equation}