Let $\sigma$ be a Borel measure on $\mathbb{R}$. Suppose we know that all Borel subsets $B \subseteq \mathbb{R}$ which contain points satisfying a certain fixed property (let's say property P) are measure $0$, that is, $\sigma(B) = 0$. My question is, can we say that the set $S_P$ of all points satisyfing property P has measure $0$? My concern is that since we do not know that $S_P$ is Borel, it could a priori be non-measurable, and I have no knowledge of non-measurable sets outside their existence (and the elementary constructions). Thanks in advance.
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