  We give a simple criterion on the set of probability tangent measures
$\mathrm{Tan}(\mu,x)$ of a positive Radon measure $\mu$, which yields lower
bounds on the Hausdorff dimension of $\mu$. As an application, we give an
elementary and purely algebraic proof of the sharp Hausdorff dimension lower
bounds for first-order linear PDE-constrained measures; bounds for closed
(measure) differential forms and normal currents are further discussed. A weak
structure theorem in the spirit of [Ann. Math. 184(3) (2016), pp. 1017-1039] is
also discussed for such measures.
