Suppose $(E_i, \mathcal E_i)$, $i = 1, \dots, n$, are measurable spaces and let $E := E_1 \times \dots \times E_n$, equipped with the product $\sigma$-algebra, denoted by $\mathcal E$.
Suppose $\psi$ is a measure on $E$, and suppose $\psi(B) > 0$ for some $B \in \mathcal E$.
(i) Are there sets $A_i \in \mathcal E_i$, $i = 1, \dots, n$, such that $A:=A_1 \times \dots \times A_n \subset B$ and $\psi(A) > 0$?
(ii) What if $\psi$ is the product measure $\psi_1 \otimes \dots \otimes \psi_n$ of some measures $\psi_i$ on $(E_i, \mathcal E_i)$?
(iii) Possibly easier, what if all $(E_i,\mathcal E_i)$ and measures $\psi_i$ are identical?