 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. 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.