{ "artifactId": "artifact.library.mathlib.fatou-lemma.v001", "candidateOnly": false, "category": "Measure theory and analysis", "claimBoundary": "This page indexes Mathlib's Fatou lemma for a sequence of measurable functions f n : α → ℝ≥0∞. It compares the lower Lebesgue integral of the pointwise liminf with the liminf of the lower Lebesgue integrals. Nonnegativity is encoded by the extended nonnegative reals, and the declaration assumes neither convergence of the sequence nor finiteness of the integrals. It does not state equality, a reverse inequality, a signed- or Bochner-integral theorem, or a quantitative rate.", "collectionSlug": "landmark-theorems-in-mathlib", "exactFormalStatement": "theorem MeasureTheory.lintegral_liminf_le {α : Type*} {m : MeasurableSpace α} {μ : Measure α} {f : ℕ → α → ℝ≥0∞} (h_meas : ∀ n, Measurable (f n)) : ∫⁻ a, liminf (fun n => f n a) atTop ∂μ ≤ liminf (fun n => ∫⁻ a, f n a ∂μ) atTop", "family": "Convergence theorems", "id": "library.mathlib.fatou-lemma.v001", "links": { "collection": "/collections/landmark-theorems-in-mathlib/", "collectionData": "/data/collections/mathlib-landmarks.json", "entry": "/library-theorems/fatou-lemma/", "entryData": "/data/library-theorems/fatou-lemma.json", "evidence": "/proofs/artifact.library.mathlib.fatou-lemma.v001.html", "evidenceData": "/data/proofs/artifact.library.mathlib.fatou-lemma.v001.evidence.json", "source": "/sources/upstream/fatou-lemma/" }, "landmark": { "completeFormalType": "theorem MeasureTheory.lintegral_liminf_le {α : Type*} {m : MeasurableSpace α} {μ : Measure α} {f : ℕ → α → ℝ≥0∞} (h_meas : ∀ n, Measurable (f n)) : ∫⁻ a, liminf (fun n => f n a) atTop ∂μ ≤ liminf (fun n => ∫⁻ a, f n a ∂μ) atTop", "concepts": [ "Fatou's lemma", "lower Lebesgue integral", "limit inferior", "extended nonnegative real functions", "monotone convergence" ], "plainLanguage": "For a sequence of measurable nonnegative extended-real functions, the integral of the pointwise lower limit cannot exceed the lower limit of the integrals.", "poster": { "alt": "An ivory measure-theory poster shows oscillating nonnegative function profiles, pointwise tail-floor samples rising toward a lower-limit profile, and nested lower-integral area bands beneath the exact Fatou inequality.", "byteSize": 2303290, "caption": "Pointwise tail infima rise to the liminf, and their lower integrals give Fatou's one-sided bound.", "description": "The poster keeps the original sequence visibly nonmonotone while gold pointwise tail-infimum profiles rise beneath it. The checked route rewrites liminf as a supremum of tail infima, passes that supremum through the lower integral, compares each tail floor with the tail integrals, and reconstructs the liminf.", "derivatives": [ { "byteSize": 89184, "height": 960, "mediaType": "image/webp", "publicPath": "/assets/library-theorems/fatou-lemma/theorem-poster-v1-640.webp", "sha256": "sha256:1ae7aa03fcfec5e7043ba7c9c00b9e3df6afd754e32e324b9dd6bafc4b3b822a", "width": 640 }, { "byteSize": 175894, "height": 1536, "mediaType": "image/webp", "publicPath": "/assets/library-theorems/fatou-lemma/theorem-poster-v1-1024.webp", "sha256": "sha256:73b819f74990b869dafb14ff9f255e7073fd0aa27bb8e1221b7aa632286b7b16", "width": 1024 } ], "generatedAt": "2026-07-25T20:33:50.511Z", "height": 1536, "mediaType": "image/png", "modelRequested": "gpt-image-2", "publicPath": "/assets/library-theorems/fatou-lemma/theorem-poster-v1.png", "sha256": "sha256:b547e805e1d25f0e8e9545df1a68067c079545903b174e356a244d2c74da04f2", "title": "Fatou's Lemma — a one-sided lower-limit bound", "width": 1024, "kind": "theorem_poster", "transcript": "MEASURE THEORY · LOWER LIMITS\nFATOU'S LEMMA\nfₙ : α → ℝ≥0∞ · each fₙ measurable\n∫⁻ a, liminfₙ fₙ(a) ∂μ ≤ liminfₙ ∫⁻ a, fₙ(a) ∂μ\nTHE POINTWISE LOWER LIMIT GIVES A ONE-SIDED INTEGRAL BOUND\nHOW THE CHECKED ROUTE MOVES\n1 · Rewrite liminf as the supremum of tail infima\n2 · Move that rising supremum through the lower integral\n3 · Bound each integrated tail infimum by the tail integrals\n4 · Reassemble the liminf of the integrals\nEXACT SCOPE\nMeasurable ℝ≥0∞-valued sequence and lower Lebesgue integrals.\nNo equality, reverse inequality, convergence, finiteness, signed-integral, or rate claim." }, "visual": { "alt": "Oscillating nonnegative function ribbons sit above nested pointwise tail-floor profiles, whose accumulated gold areas form a one-sided lower-limit comparison.", "byteSize": 3069681, "caption": "Rising pointwise tail infima reveal the liminf profile that supplies Fatou's lower-integral bound.", "description": "A nonmonotone sequence of engraved green and cobalt profiles is sampled point by point. Nested gold tail floors rise beneath it toward the lower-limit profile, and their accumulated areas remain visually distinct from the open-ended later-integral stack.", "derivatives": [ { "byteSize": 55180, "height": 427, "mediaType": "image/webp", "publicPath": "/assets/library-theorems/fatou-lemma/theorem-schematic-v1-640.webp", "sha256": "sha256:462f9ccaae1c3a354dd5ed6a6e205720153ece84c9d6f87e29933877862cdc25", "width": 640 }, { "byteSize": 197576, "height": 800, "mediaType": "image/webp", "publicPath": "/assets/library-theorems/fatou-lemma/theorem-schematic-v1-1200.webp", "sha256": "sha256:449bcf21cd98763a1f0d87c0f6998468957a72859d51a55d60b308eafeaca9ce", "width": 1200 } ], "generatedAt": "2026-07-25T20:33:50.511Z", "height": 1024, "mediaType": "image/png", "modelRequested": "gpt-image-2", "publicPath": "/assets/library-theorems/fatou-lemma/theorem-schematic-v1.png", "sha256": "sha256:9fbf800ffc84217916db7a39cf88ebd1804b62275fb2629e1a946733ea5b8519", "title": "Tail floors rise to the pointwise liminf", "width": 1536 }, "walkthrough": null, "whyLandmark": "Fatou's lemma is a foundational lower-semicontinuity principle for integration. It controls the integral of a pointwise lower limit without requiring the sequence itself to converge and is a standard gateway to stronger convergence theorems." }, "nonClaims": [ "The selected declaration is one-sided: it does not assert equality or the reverse inequality.", "It does not assume that the sequence converges, is monotone, or has finite integrals.", "It is the ℝ≥0∞-valued lower-Lebesgue-integral form, not a signed- or Bochner-integral statement.", "Proof Atlas did not originate Fatou's lemma or Mathlib's declaration.", "The generated explanation and visuals are not proof evidence." ], "publicExportEligible": true, "relationship": "upstream_direct", "schemaVersion": "library-theorem-entry-data.v1", "statementId": "statement.library.mathlib.fatou-lemma.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": "For measurable ℝ≥0∞-valued functions, the lower integral of the pointwise liminf is at most the liminf of the lower integrals.", "targetId": "target.library.mathlib.fatou-lemma.v001", "title": "Fatou's Lemma", "upstreamOrigin": { "declarationName": "MeasureTheory.lintegral_liminf_le", "immutableSourceUrl": "https://github.com/leanprover-community/mathlib4/blob/5e932f97dd25535344f80f9dd8da3aab83df0fe6/Mathlib/MeasureTheory/Integral/Lebesgue/Add.lean#L214", "packageName": "mathlib", "repositoryUrl": "https://github.com/leanprover-community/mathlib4", "sourceArtifactHash": "sha256:992bbef3b374a1bc0d2677d91e48195121d4adc86d08ccae8c0f735b7a596186", "sourceByteLength": 22819, "sourceCommit": "5e932f97dd25535344f80f9dd8da3aab83df0fe6", "sourceFile": "Mathlib/MeasureTheory/Integral/Lebesgue/Add.lean", "sourceLine": 214, "verificationKind": "git_worktree" }, "publicPresentationReview": { "status": "reviewed" } }