Unique continuation for locally uniformly distributed measures

In this note we show that the support of a locally $k$-uniform measure in ${\mathbb{R}}^{n+1}$ satisfies a kind of unique continuation property. As a consequence, we show that locally uniformly distributed measures satisfy a weaker unique continuation property. This continues work of Kirchheim and Preiss (Math. Scand. 2002) and David, Kenig and Toro (Comm. Pure Appl. Math. 2001) and lends additional evidence to the conjecture proposed by Kowalski and Preiss (J. Reine Angew. Math. 1987) that each connected component of the support of a locally $n$-uniform measure in ${\mathbb{R}}^{n+1}$ is contained in the zero set of a quadratic polynomial.