The Minkowski content is (generally) not a measure. In particular, the m-dimensional Minkowski content in Rn is not a measure unless m = 0, in which case it is the . Indeed, clearly the Minkowski content assigns the same value to the set A as well as its closure.
counting measure
If A is a closed m- in Rn, given as the image of a bounded set from Rm under a Lipschitz function, then the m-dimensional Minkowski content of A exists, and is equal to the m-dimensional Hausdorff measure of A.[3]
rectifiable set
Gaussian isoperimetric inequality
Geometric measure theory
Isoperimetric inequality in higher dimensions
Minkowski–Bouligand dimension
Krantz, Steven G.; (1999), The geometry of domains in space, Birkhäuser Advanced Texts: Basler Lehrbücher, Boston, MA: Birkhäuser Boston, Inc., ISBN 0-8176-4097-5, MR 1730695.