= Solution
A <Sierpiński set> is an uncountable <subset> $S$ of $\mathbb R$ such that $S\cap N$ is countable for every <set> $N$ of <Lebesgue measure> zero. Equivalently it is enough to test <Borel null sets>, because every <null set> is contained in a Borel <null set>. \b[Uncountability and countable intersection with every <null set> are both required.] This is the measure analogue of the category-based <Luzin set> condition; it does not say that $S$ itself is measurable.
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