Stratification of the single blow-up set for Radon measures

We show that the set of points where the blow-up, in the sense of Preiss, of a signed Radon measure on ${\mathbb{R}}^n$ is unique and its invariant subspace has dimension $k$ is $k$-rectifiable. As simple applications, we obtain a rectifiability criterion for signed Radon measures and the extension of a result, due to Mattila, on measures having unique blow-up almost everywhere.