For a set $X$, let $\mathcal P(X)$ denote its power set and let $\mathcal P(X)\otimes\mathcal P(X)$ denote the product $\sigma$-algebra in $X^2$. When $|X|\leq\aleph_0$ then $\mathcal P(X)\otimes\mathcal P(X)=\mathcal P(X^2)$ but when $|X|>2^{\aleph_0}$ this equality is known to fail. What happens when $\aleph_0<|X|\leq 2^{\aleph_0}$?