If $\mu$ and $\nu$ are measures on $X$ and $Y$, is there an example of a set $E\subset X\times Y$ such that $E_x,E^y$ are measurable for all $(x,y)$ but $E$ is not measurable with respect to $\mu\times\nu$?
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Potential duplicate: http://math.stackexchange.com/questions/590251/non-measurable-set-in-product-sigma-algebra-s-t-every-section-is-measurable?rq=1 – Math1000 Jun 28 '16 at 18:32