Take a measure on the real line. Shift it by a small distance h in both directions, superpose the two copies, and compare the result to the original using the Wasserstein metric. The distance between original and perturbed measure is bounded by h — the perturbation can't move mass farther than it shifts it. Most measures change by less than this bound. The question: which measures saturate it?
Aussedat characterizes the critically unstable measures — those that achieve the maximum rate of change as h approaches zero. They are precisely the measures that concentrate their mass on porous sets: sets with holes at every scale, where every interval contains gaps proportional to its length. The porosity is the geometric property that makes the measure maximally sensitive to shifts.
The intuition: a measure spread smoothly along an interval is stable under shifts because the shifted copies overlap substantially with the original. A measure concentrated on a porous set is unstable because the gaps at every scale mean the shifted copies fill different holes, producing maximal displacement at every resolution. The porosity ensures that no matter how small the shift, the copies always find new space to occupy.
The through-claim is about the geometry of sensitivity. Stability under perturbation is a property of how mass is distributed in space. Smooth distributions are stable. Fractal-like distributions on porous sets are maximally unstable. The connection between porosity and Wasserstein sensitivity reveals that sensitivity to small perturbations is not about the total mass or its mean location but about the fine structure of how mass fills space. The gaps are what makes the measure fragile.