{ "artifactId": "artifact.library.mathlib.radon-nikodym-theorem.v001", "candidateOnly": false, "category": "Measure theory and analysis", "claimBoundary": "This page indexes Mathlib's positive-measure Radon–Nikodym theorem. For measures μ and ν on one measurable space, it assumes HaveLebesgueDecomposition μ ν and states the exact equivalence μ ≪ ν ↔ ν.withDensity (rnDeriv μ ν) = μ. The density is ENNReal-valued. The selected declaration is not the signed-measure theorem, not a vector-measure theorem, and not an assumption-free existence claim for arbitrary measure pairs.", "collectionSlug": "landmark-theorems-in-mathlib", "exactFormalStatement": "theorem MeasureTheory.Measure.absolutelyContinuous_iff_withDensity_rnDeriv_eq {α : Type*} {m : MeasurableSpace α} {μ ν : Measure α} [HaveLebesgueDecomposition μ ν] : μ ≪ ν ↔ ν.withDensity (rnDeriv μ ν) = μ", "family": "Measure decomposition and density", "id": "library.mathlib.radon-nikodym-theorem.v001", "links": { "collection": "/collections/landmark-theorems-in-mathlib/", "collectionData": "/data/collections/mathlib-landmarks.json", "entry": "/library-theorems/radon-nikodym-theorem/", "entryData": "/data/library-theorems/radon-nikodym-theorem.json", "evidence": "/proofs/artifact.library.mathlib.radon-nikodym-theorem.v001.html", "evidenceData": "/data/proofs/artifact.library.mathlib.radon-nikodym-theorem.v001.evidence.json", "source": "/sources/upstream/radon-nikodym-theorem/" }, "landmark": { "completeFormalType": "theorem MeasureTheory.Measure.absolutelyContinuous_iff_withDensity_rnDeriv_eq {α : Type*} {m : MeasurableSpace α} {μ ν : Measure α} [HaveLebesgueDecomposition μ ν] : μ ≪ ν ↔ ν.withDensity (rnDeriv μ ν) = μ", "concepts": [ "absolute continuity of measures", "Radon–Nikodym derivative", "measure density", "Lebesgue decomposition", "ENNReal-valued functions" ], "plainLanguage": "When μ assigns zero mass wherever ν does, the Radon–Nikodym derivative acts as a nonnegative density that reweights ν to recover all of μ exactly—and the reconstruction equality also implies that absolute continuity.", "poster": { "alt": "An ivory analysis poster layers a forest-green measure ν, an antique-gold Radon–Nikodym density, and the reconstructed cobalt measure μ above the exact biconditional and positive-measure scope.", "byteSize": 2900445, "caption": "Assuming the recorded Lebesgue decomposition, absolute continuity is equivalent to exact reconstruction by the Radon–Nikodym density.", "description": "The poster presents the assumption HaveLebesgueDecomposition μ ν, then aligns a green ν base, a variable gold density layer, and a blue μ field. The central row states μ ≪ ν if and only if ν.withDensity (rnDeriv μ ν) = μ. A muted red singular fragment dissolves before a footer limits the result to positive measures and excludes signed- and vector-measure claims.", "derivatives": [ { "byteSize": 124374, "height": 960, "mediaType": "image/webp", "publicPath": "/assets/library-theorems/radon-nikodym-theorem/theorem-poster-v1-640.webp", "sha256": "sha256:7ce8799a78165362038bcb8894b10b8a4f9b5c046d3109febbd64501ba1616f5", "width": 640 }, { "byteSize": 295362, "height": 1536, "mediaType": "image/webp", "publicPath": "/assets/library-theorems/radon-nikodym-theorem/theorem-poster-v1-1024.webp", "sha256": "sha256:c066e049d4ddc6c779bae07074abcb88bcd705cf797fe6d85e979300b07e9e7d", "width": 1024 } ], "generatedAt": "2026-07-25T20:33:59.710835126Z", "height": 1536, "mediaType": "image/png", "modelRequested": "gpt-image-2", "publicPath": "/assets/library-theorems/radon-nikodym-theorem/theorem-poster-v1.png", "sha256": "sha256:fb113fe0c6d594e67963a10a29e0ec1145698e7f630a3b0cecd1842de6dbff49", "title": "Radon–Nikodym Theorem at a glance", "width": 1024, "kind": "theorem_poster", "transcript": "MEASURE THEORY · DENSITY REPRESENTATION\nRADON–NIKODYM THEOREM\nMeasures μ and ν\nAssume HaveLebesgueDecomposition μ ν\nμ ≪ ν\nIF AND ONLY IF\nν.withDensity (rnDeriv μ ν) = μ\nNO SINGULAR PART REMAINS\nEXACT SCOPE\nPositive measures. ENNReal-valued density.\nNot the signed- or vector-measure theorem." }, "visual": { "alt": "A text-free layered measure field shows a variable gold density over a green base resolving into a complete blue measure, while a separate red singular shard dissolves to zero.", "byteSize": 3202408, "caption": "When no singular mass remains, the Radon–Nikodym density over ν reconstructs all of μ.", "description": "A forest-green cellular base supports an antique-gold variable density canopy and a cobalt reconstructed field. Balanced opposing arcs make the relationship reversible. In a secondary lower stratum, a muted vermilion singular shard breaks apart and vanishes before the density reconstruction completes.", "derivatives": [ { "byteSize": 65442, "height": 427, "mediaType": "image/webp", "publicPath": "/assets/library-theorems/radon-nikodym-theorem/theorem-schematic-v1-640.webp", "sha256": "sha256:312ba3dd463e2dc98f478d446f6b51f3729c63c46edd306cc2313b4bca2c5a31", "width": 640 }, { "byteSize": 250626, "height": 800, "mediaType": "image/webp", "publicPath": "/assets/library-theorems/radon-nikodym-theorem/theorem-schematic-v1-1200.webp", "sha256": "sha256:9141b0c665e927bf302b8f1d1711bb42324442a81fbf5f2fe36533fe264cb965", "width": 1200 } ], "generatedAt": "2026-07-25T20:33:59.710835126Z", "height": 1024, "mediaType": "image/png", "modelRequested": "gpt-image-2", "publicPath": "/assets/library-theorems/radon-nikodym-theorem/theorem-schematic-v1.png", "sha256": "sha256:e52dfe8e6c7566b9d16f53b747f7b60986b17b1b62b5553610f1721d43fd6d23", "title": "Radon–Nikodym density reconstruction schematic", "width": 1536 }, "walkthrough": null, "whyLandmark": "The Radon–Nikodym theorem is the central representation principle for absolutely continuous measures. It turns a qualitative relation between null sets into an exact density formula and underlies foundational constructions throughout analysis and probability." }, "nonClaims": [ "Proof Atlas did not originate the Radon–Nikodym theorem or Mathlib's declaration.", "The selected declaration requires HaveLebesgueDecomposition μ ν and concerns positive measures with an ENNReal-valued density, not the signed- or vector-measure variants.", "The generated explanation and visuals are not proof evidence." ], "publicExportEligible": true, "relationship": "upstream_direct", "schemaVersion": "library-theorem-entry-data.v1", "statementId": "statement.library.mathlib.radon-nikodym-theorem.v001", "statusAxes": { "acceptedAtlasResult": "Not recorded for the preferred artifact", "locallyReproduced": "Exact upstream declaration replayed", "reviewedPage": "Current public presentation reviewed", "upstreamIndexed": "Pinned source bytes verified locally" }, "summary": "Under the required Lebesgue decomposition, μ is absolutely continuous with respect to ν exactly when weighting ν by rnDeriv μ ν reconstructs μ.", "targetId": "target.library.mathlib.radon-nikodym-theorem.v001", "title": "Radon–Nikodym Theorem", "upstreamOrigin": { "declarationName": "MeasureTheory.Measure.absolutelyContinuous_iff_withDensity_rnDeriv_eq", "immutableSourceUrl": "https://github.com/leanprover-community/mathlib4/blob/5e932f97dd25535344f80f9dd8da3aab83df0fe6/Mathlib/MeasureTheory/Measure/Decomposition/RadonNikodym.lean#L71", "packageName": "mathlib", "repositoryUrl": "https://github.com/leanprover-community/mathlib4", "sourceArtifactHash": "sha256:2807db6d77bb145e00e953f9b425e1775794aa68b41a01dac46a3fb97e4dd970", "sourceByteLength": 36955, "sourceCommit": "5e932f97dd25535344f80f9dd8da3aab83df0fe6", "sourceFile": "Mathlib/MeasureTheory/Measure/Decomposition/RadonNikodym.lean", "sourceLine": 71, "verificationKind": "git_worktree" }, "publicPresentationReview": { "status": "reviewed" } }