Measurable set with respect to a product measure

Question

Let $(X,\mathcal{A},\mu)$ and $(Y,\mathcal{B}, \nu)$ be two measure spaces. Suppose that $A\times B\subset X\times Y$ is measurable with respect to the product measure $\mu\times\nu$ and $B\in\mathcal{B}$. Is $A\in\mathcal{A}$?

Answer 1

If $\mathcal A =\mathcal B=$ Borel sigma algebra of $\mathbb R$, $\mu=\nu=$ Lebesgue measure and $A$ is a Lebesgue measurable set in $\mathbb R$ which is not a Borel set then $A\times \mathbb R$ is measurable with respect to the product measure but $A \notin \mathcal A$. If you really meant that $A\times B$ is $\mathcal A \times\mathcal B$ then it does follow that $A \in \mathcal A$. This is part of Fubini's Theorem.