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 90(1): 152-160, 2002) and David, Kenig and Toro (Comm Pure Appl Math 54(4): 385-449, 2001) and lends additional evidence to the conjecture proposed by Kowalski and Preiss (J Reine Angew Math 379: 115-151, 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.
