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.
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