{
  "schema_version": 1,
  "problem_number": "AMR-074-0117",
  "title": "The Lebesgue Differentiation Theorem over RCA0: On Question 17 of Montalbán's Open Questions in Reverse Mathematics",
  "author": {
    "name": "Alper Ferudun",
    "affiliation": "Mercury Software GmbH",
    "url": "https://github.com/AlperTheKing"
  },
  "abstract": "Question 17 of Montalbán's survey of open questions in reverse mathematics (Bull. Symbolic Logic 17 (2011)) asks whether the Lebesgue differentiation theorem (LDT) implies WWKL0 over RCA0; Simpson had observed, using Pathak's work, that LDT is provable in WWKL0. The question does not fix how \"for almost every x\" and the measure of an open set are formalised, and the answer depends on this. We prove in RCA0: for every L1 function f = (p_n) on [0,1]^d, given by rational polynomials with ‖p_n − p_(n+1)‖₁ ≤ 2^(−n), there are open sets U_N, uniformly Δ⁰₁ in f, of measure at most ε_N → 0 (for the extensional measure of Brown, Giusto and Simpson and for the measure μ̄ of Nies, Triplett and Yokoyama), such that every point outside some U_N is a Lebesgue point of f with f(x) = lim p_n(x), and every box of measure > ε_N meets the complement of U_N. Hence RCA0 proves LDT under the null-cover reading of Yu and of Brown, Giusto and Simpson, under the reading that Lebesgue points exist and are dense, and under the randomness reading; under these readings LDT does not imply WWKL0. Under the outer reading of Nies, Triplett and Yokoyama, LDT is equivalent to WWKL0; the reversal uses their open set of measure < 1 whose complement consists of dyadic rationals, and it reflects the behaviour of measure in RCA0, not differentiation. We judge the null-cover reading to be the intended one, so that the answer is no. The method makes the maximal-function estimates of Pathak, Rojas and Simpson finite. Yu proved in RCA0 that the values p_n(x) converge outside a Martin-Löf null set; our results are close to known techniques and may be known to experts, but we found no printed answer to the question. This is an unrefereed note.",
  "result_type": "COMPLETE_PROOF",
  "categories": [
    "math.LO"
  ],
  "keywords": [
    "reverse mathematics",
    "Lebesgue differentiation theorem",
    "weak weak König's lemma",
    "WWKL0",
    "RCA0",
    "Lebesgue points",
    "Martin-Löf randomness",
    "Schnorr randomness",
    "measure of open sets",
    "Montalbán's open questions in reverse mathematics",
    "UnsolvedMath",
    "AMR-074-0117",
    "math.LO",
    "open mathematics"
  ],
  "manuscript_version_date": "2026-10-18",
  "publication_date": "2026-10-18",
  "publication_date_kind": "first public online release",
  "version": "1.0",
  "date_modified": "2026-10-18",
  "presentation_revision_only": false,
  "doi_archived_file_version": "1.0",
  "status": "unrefereed preprint",
  "canonical_url": "https://eulersolve.org/papers/amr-074-0117/",
  "pdf_url": "https://eulersolve.org/papers/amr-074-0117/paper.pdf?v=ceb609dcd5a8",
  "doi": "10.5281/zenodo.23455507",
  "zenodo_record_url": "https://zenodo.org/records/23455507",
  "license": "https://creativecommons.org/licenses/by/4.0/",
  "scope_caveat": "The question does not fix how 'almost every' is formalised: under the null-cover, existence and randomness readings RCA0 proves the theorem (so it does not imply WWKL0); under the outer reading of Nies, Triplett and Yokoyama it is equivalent to WWKL0. The choice of the intended reading is a judgement. Close to known techniques; no priority is claimed.",
  "files": {
    "paper.pdf": {
      "sha256": "ceb609dcd5a8517f59a09002f1f018c6bb515e55574f3393b5e9565928bae846"
    },
    "source.zip": {
      "sha256": "85586efbdc5924f5aa4bc2f5c7b27eb05622a9b6cf25257722c959a2da2b3a26"
    },
    "verification_report.md": {
      "sha256": "99a09bf30f4f4390e4a27f7941a8ad6c72f107dad9b0b971e456df815b1587c8"
    }
  },
  "ai_use_disclosure": "AI-assisted tools supported research, computation, proof development, and manuscript preparation. The author remains responsible for all claims and the final text."
}