{ "artifactId": "artifact.library.mathlib.monotone-convergence-theorem.v001", "candidateOnly": false, "category": "Measure theory and analysis", "claimBoundary": "This page indexes Mathlib's monotone convergence theorem for a natural-number-indexed sequence f : ℕ → α → ℝ≥0∞ whose terms are measurable and which is monotone in the pointwise function order. It concludes that the lintegral of the pointwise supremum equals the supremum of the lintegrals. Nonnegativity is encoded by ℝ≥0∞, so no integrability or finiteness hypothesis is required and either side may be ∞. The selected declaration does not state the nearby almost-everywhere-measurable/almost-everywhere-monotone variant, a signed or Bochner-integral theorem, convergence for a nonmonotone sequence, or a quantitative rate.", "collectionSlug": "landmark-theorems-in-mathlib", "exactFormalStatement": "theorem MeasureTheory.lintegral_iSup {α : Type*} {m : MeasurableSpace α} {μ : Measure α} {f : ℕ → α → ℝ≥0∞} (hf : ∀ n, Measurable (f n)) (h_mono : Monotone f) : ∫⁻ a, ⨆ n, f n a ∂μ = ⨆ n, ∫⁻ a, f n a ∂μ", "family": "Convergence theorems", "id": "library.mathlib.monotone-convergence-theorem.v001", "links": { "collection": "/collections/landmark-theorems-in-mathlib/", "collectionData": "/data/collections/mathlib-landmarks.json", "entry": "/library-theorems/monotone-convergence-theorem/", "entryData": "/data/library-theorems/monotone-convergence-theorem.json", "evidence": "/proofs/artifact.library.mathlib.monotone-convergence-theorem.v001.html", "evidenceData": "/data/proofs/artifact.library.mathlib.monotone-convergence-theorem.v001.evidence.json", "source": "/sources/upstream/monotone-convergence-theorem/" }, "landmark": { "completeFormalType": "theorem MeasureTheory.lintegral_iSup {α : Type*} {m : MeasurableSpace α} {μ : Measure α} {f : ℕ → α → ℝ≥0∞} (hf : ∀ n, Measurable (f n)) (h_mono : Monotone f) : ∫⁻ a, ⨆ n, f n a ∂μ = ⨆ n, ∫⁻ a, f n a ∂μ", "concepts": [ "monotone convergence", "Lebesgue integration", "nonnegative extended reals", "pointwise suprema", "measurable functions", "lintegrals" ], "plainLanguage": "If measurable nonnegative functions grow pointwise with the sequence index, then integrating their pointwise supremum gives the same extended value as taking the supremum of their integrals.", "poster": { "alt": "An ivory measure-theory poster states the measurable pointwise-monotone hypotheses and the equality between the lintegral of a pointwise supremum and the supremum of the lintegrals.", "byteSize": 2618564, "caption": "For measurable ℝ≥0∞-valued functions increasing in pointwise order, lintegral commutes with the pointwise supremum.", "description": "The poster sets four nested green profiles above growing cobalt mass basins and centers the exact nonnegative extended-real lintegral equality. Its footer states the natural-number sequence, absence of integrability or finiteness hypotheses, and possible infinite common value.", "derivatives": [ { "byteSize": 83872, "height": 960, "mediaType": "image/webp", "publicPath": "/assets/library-theorems/monotone-convergence-theorem/theorem-poster-v1-640.webp", "sha256": "sha256:9a9ff187aa6ed73f3dd3cc295b20d8cd7998d09adc17b45a4f2fe3629ed4cd74", "width": 640 }, { "byteSize": 197070, "height": 1536, "mediaType": "image/webp", "publicPath": "/assets/library-theorems/monotone-convergence-theorem/theorem-poster-v1-1024.webp", "sha256": "sha256:05ae50c5f76a3f75509ef270cdf80079726f071ad4d38328b90eb2d23321797c", "width": 1024 } ], "generatedAt": "2026-07-25T16:27:31-04:00", "height": 1536, "mediaType": "image/png", "modelRequested": "gpt-image-2", "publicPath": "/assets/library-theorems/monotone-convergence-theorem/theorem-poster-v1.png", "sha256": "sha256:437f36ec301a84395103cf03af1be8ef9d38530373c2757a29c0a3629c63675f", "title": "Monotone Convergence Theorem at a glance", "width": 1024, "kind": "theorem_poster", "transcript": "MEASURE THEORY · MONOTONE LIMITS\nMONOTONE CONVERGENCE THEOREM\nfₙ : α → ℝ≥0∞\nfₙ measurable\nfₙ pointwise monotone\n∫⁻ (supₙ fₙ) dμ = supₙ ∫⁻ fₙ dμ\nASCENDING FUNCTIONS · ASCENDING MASS\nEXACT SCOPE\nNatural-number sequences · nonnegative extended-real lintegrals\nNo integrability or finiteness hypothesis. The common value may be ∞." }, "visual": { "alt": "A text-free engraved landscape shows nested nonnegative function profiles rising pointwise while connected accumulation basins grow toward the same terminal endpoint.", "byteSize": 3217023, "caption": "A pointwise-monotone sequence builds its supremum while the corresponding nonnegative lintegrals rise to the same extended-real value.", "description": "Five nested emerald profiles rise everywhere toward a cobalt terminal silhouette over a shared domain. Fine gold fibers connect them to successively larger blue accumulation basins below, and the open, dissolving upper edge leaves room for the extended value infinity without asserting that it occurs.", "derivatives": [ { "byteSize": 40168, "height": 427, "mediaType": "image/webp", "publicPath": "/assets/library-theorems/monotone-convergence-theorem/theorem-schematic-v1-640.webp", "sha256": "sha256:8ed704f6ab1f5ef74b597c56d2aa8694d9837f78569399413e5c63205ef94149", "width": 640 }, { "byteSize": 170020, "height": 800, "mediaType": "image/webp", "publicPath": "/assets/library-theorems/monotone-convergence-theorem/theorem-schematic-v1-1200.webp", "sha256": "sha256:6855e5527535a8826b8274a1b178879fd9222aa3a00aa2cedcaf6a9846c6834b", "width": 1200 } ], "generatedAt": "2026-07-25T16:27:31-04:00", "height": 1024, "mediaType": "image/png", "modelRequested": "gpt-image-2", "publicPath": "/assets/library-theorems/monotone-convergence-theorem/theorem-schematic-v1.png", "sha256": "sha256:c53a6f17166351c1b98581d1e179364a7b1e8d31fd5e5ee47b93c685e955a697", "title": "Ascending profiles and ascending nonnegative mass", "width": 1536 }, "walkthrough": null, "whyLandmark": "The monotone convergence theorem is a foundational limit-interchange principle in measure theory. It allows nonnegative mass to be assembled through an increasing sequence without an integrable dominating function and underlies the construction and structural properties of the Lebesgue integral." }, "nonClaims": [ "Proof Atlas did not originate the monotone convergence theorem or Mathlib's declaration.", "The selected declaration requires ordinary measurability of every term and pointwise monotonicity in the function order, not merely almost-everywhere versions.", "The conclusion concerns ℝ≥0∞-valued lintegrals and permits the common value ∞; it is not a signed or Bochner-integral statement.", "The generated explanation and visuals are not proof evidence." ], "publicExportEligible": true, "relationship": "upstream_direct", "schemaVersion": "library-theorem-entry-data.v1", "statementId": "statement.library.mathlib.monotone-convergence-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": "The nonnegative extended-real lintegral of the pointwise supremum of a measurable monotone sequence equals the supremum of its lintegrals.", "targetId": "target.library.mathlib.monotone-convergence-theorem.v001", "title": "Monotone Convergence Theorem", "upstreamOrigin": { "declarationName": "MeasureTheory.lintegral_iSup", "immutableSourceUrl": "https://github.com/leanprover-community/mathlib4/blob/5e932f97dd25535344f80f9dd8da3aab83df0fe6/Mathlib/MeasureTheory/Integral/Lebesgue/Add.lean#L34", "packageName": "mathlib", "repositoryUrl": "https://github.com/leanprover-community/mathlib4", "sourceArtifactHash": "sha256:992bbef3b374a1bc0d2677d91e48195121d4adc86d08ccae8c0f735b7a596186", "sourceByteLength": 22819, "sourceCommit": "5e932f97dd25535344f80f9dd8da3aab83df0fe6", "sourceFile": "Mathlib/MeasureTheory/Integral/Lebesgue/Add.lean", "sourceLine": 34, "verificationKind": "git_worktree" }, "publicPresentationReview": { "status": "reviewed" } }