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A Borel set of Lebesgue measure zero. Every Lebesgue-null set is contained in a Borel null set, by outer regularity; consequently testing all Borel null sets suffices in the definition of a Sierpiński set.

An uncountable subset of the real numbers with countable intersection with every meagre set. This is the category counterpart of a Sierpiński set.

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. 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.