An ergodic Lebesgue differentiation theorem

The paper proves the two-parameter a.e. convergence lim_{r↓0,k→∞} (1/µ(B(x,r)))∫_{B(x,r)} A_k f dµ = E[f|I](x) for f∈L^p, p>1, on metric Borel probability spaces with the Hardy–Littlewood property, via a dense L∞-set plus a maximal-inequality/Banach-principle argument (Lemma 4, Lemma 6, Lemma 7, and the proof of Theorem B) . The candidate solution gives a different, valid proof: it reduces to Birkhoff + Lebesgue differentiation and then builds a uniform-in-(r,k) modulus on a full-measure set using Egorov, Lusin, and HL maximal control. This is compatible with the paper’s assumptions and conclusion (indeed slightly stronger in uniformity on a full-measure set), while the candidate’s remark claiming an L^1 extension contradicts the paper’s open Question 10 (the L^1 case without rate conditions is explicitly left open) .