Indistinguishability of cells for the ideal Poisson Voronoi tessellation

The paper proves indistinguishability of IPVT cells via Meyerovitch’s criterion: if G acts ergodically on X×M(X), then the Poisson process on X admits no nontrivial deterministic equivariant thinning; they verify the needed ergodicity (via mixing of the action on M(Z)) and conclude for X=Z. The candidate provides an alternative proof using Palm/Mecke calculus, identifies Z with a Maharam/homogeneous space, and applies the Mautner/Howe–Moore phenomenon to force the Palm retention indicator to be constant, hence the thinning is a.s. full or empty. The two arguments are logically consistent; the paper’s Lemma 6 is phrased for G×M(Z), where a short clarification that ergodicity on Z×M(Z) follows (e.g., by ergodic G↷Z and mixing on M(Z)) would make the application to Theorem 2 fully explicit.