THE DISINTEGRATION OF MEASURES WITH PERIODIC REPETITIVE PATTERN

The paper proves that any limit measure with periodic repetitive pattern has atoms in its fiber disintegration, and, if ergodic, the disintegration is atomic almost everywhere (Theorem 2) using nested horizontal strips An with a uniform lower bound on their mass, derived from the block–tail combinatorics and the summability of λn, and then passing to U = ∩n(Σk × An) with μ(U) ≥ ρ > 0 to force atoms in fibers . The candidate solution mimics the block combinatorics but makes two critical errors: (i) it replaces the paper’s safe product ρ = ∏n(1+λn)−1 by δ = ∏n(1−λn), which need not be positive unless every λn<1, a condition not assumed; and (ii) it misapplies continuity from above to conclude μ(Σk×{xℓ}) ≥ δ/πm from a lower bound proven only for the larger sets Σk×Im,ℓ, without establishing a uniform lower bound for the decreasing family Σk×In,ℓ. Consequently, the model’s argument does not validly produce atoms, whereas the paper’s proof does.