# measure of the boundary of the support

Let $\mu$ be a Borel probability measure on $\mathbb R^d$. Does the boundary of the support of $\mu$ have measure zero, i.e. do we have $$\mu(\partial(\text{supp}\mu))=0,$$ where we define the support of $\mu$ as the smallest closed set such that its complement has $\mu$-measure zero?

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No; for instance, consider the case that $\mu$ is a point mass.