Random Expansive Measures

The paper’s Theorem A asserts that if an ergodic F-invariant measure μ has positive random (fiber) metric entropy h_μ(F|θ)>0, then for P-a.e. w every w-stable class has μ_w-measure zero; this statement and its proof route are clearly presented in the uploaded PDF (Theorem A and its proof via Lemmas 3.3 and 3.4) . The paper proves (i) positive entropy ⇒ positively random μ-expansive via a random Brin–Katok formula, then (ii) positively random μ-expansive ⇒ stable classes have zero conditional measure; combining yields Theorem A. The candidate solution proves the same conclusion directly from the random Brin–Katok local entropy formula by showing exponential decay of forward Bowen balls μ_w(B_n^w(p,ε)), deducing the tail intersections have μ_w-measure zero, and then representing W^s_w(p) as a tail of forward Bowen balls; this is a sound but slightly different route. Minor issues: the paper states Lemma 3.4 under 0 < h_μ(F) < ∞ but Theorem A cites only h_μ(F|θ)>0; however, the argument naturally requires finiteness (as used in Lemma 3.4) . The model slightly overclaims that the tail-intersection has zero measure for every ε>0 (one only needs a sequence ε→0). Aside from these small points, both arguments are correct and reach the same result.