Does Hausdorff measure have a physical meaning?

Question

Although Hausdorff measure provides a reasonable way to approach general metric space including fractal geometry, I do not really realize how they actually get meanings in the physical world.

May I ask for some examples where the connection between Hausdorff measure and physical phenomenon has been observed?

I meant some concrete properties like mass, conductivity, or probability distribution associated with Hausdorff measure. Any measures having such realization other than Lebesgue and probability measures will be also helpful. Thank you.

Answer 1

The problem is that people don't believe that fractal geometry has application in real world constructs, but that isn't the case. Consider the "length" of coastlines -- different texts have vastly different measurements for the length of a coastline depending on the scale at which it was measured, because coastlines are somewhat fractal in nature, even though they are real physical objects. The west coast of Ireland has been calculated to have a Hausdorff dimension of roughly 1.26, while the east coast of Ireland has a Hausdorff dimension of roughly 1.10. This tells you that (a) it is almost pointless to measure the linear distance of these coastlines, since they are fractal and will measure "longer" the smaller the scale you use, but also that (b) the west coast is "more crinkly" than the east coast. Hausdorff dimension gives a way to measure concretely how much more "folded in on itself" an object, curve, or other things to be measured might be, including real world objects.

Answer 2

$n$-dimensional Hausdorff measure on $\mathbb{R}^n$ coincides with Lebesgue measure. For $m < n$, $m$-dimensional Hausdorff measure on $\mathbb{R}^n$ coincides with the standard surface measure on embedded manifolds $M \subset \mathbb{R}^n$. The standard surface measure is the one that defines formulas for length of curve, area of surface, etc. that you compute in calculus class.