Let $(X, \mathcal{J})$ and $(Y, \mathcal{F})$ be two measure spaces. Let us assume that $J$ is a collection of subsets of $X$ which generates $\mathcal{J}$, i.e. $\sigma(J)=\mathcal{J}$. Similarly, assume $\sigma(F)=\mathcal{F}$. Is it always true that $ \sigma(J)\times\sigma(F)=\sigma(J\times F)? $ Here, $J\times F$ is the set of all cartesian products of sets in $J$ with sets in $F$.
Please note: in the above expression the two product signs mean different things. On the left I am considering the product sigma algebra $\mathcal{J}\times \mathcal{F}$, while on the right I am considering the $\sigma$-algebra generated by the cartesian products of my ``elementary sets.''