= Solution
\b[False.] Take $E=\{0\}$ and any <non-measurable set> $A\subseteq\mathbb R$. The vertical line $\{0\}\times\mathbb R$ is a planar <Lebesgue null set>: each segment $\{0\}\times[-N,N]$ can be enclosed in rectangles of arbitrarily small area, and the line is a countable union of these segments. Consequently its <subset> $E\times A$ is a <Lebesgue measurable set> of planar measure zero, by <Completeness of Lebesgue measure>.
A nonempty factor can therefore have too little <Lebesgue measure> to transmit the other factor's failure of measurability. Its exceptional vertical section at $x=0$ is consistent with <Lebesgue measurable sections> being guaranteed only <almost everywhere>.
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