{
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  "claimBoundary": "This page indexes Mathlib's sequential dominated-convergence theorem for Bochner integrals of real-normed-space-valued functions. It assumes almost-everywhere strong measurability of every term, one integrable real-valued bound dominating every norm almost everywhere, and almost-everywhere pointwise convergence. The declaration does not assume completeness; its substantive dominated-convergence content is the complete-valued case, while Mathlib's Bochner integral is definitionally zero for incomplete codomains. It does not state monotone convergence, uniform convergence, pointwise-everywhere convergence, or a quantitative rate.",
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  "exactFormalStatement": "theorem MeasureTheory.tendsto_integral_of_dominated_convergence {α G : Type*} [NormedAddCommGroup G] [NormedSpace ℝ G] {m : MeasurableSpace α} {μ : Measure α} {F : ℕ → α → G} {f : α → G} (bound : α → ℝ) (F_measurable : ∀ n, AEStronglyMeasurable (F n) μ) (bound_integrable : Integrable bound μ) (h_bound : ∀ n, ∀ᵐ a ∂μ, ‖F n a‖ ≤ bound a) (h_lim : ∀ᵐ a ∂μ, Filter.Tendsto (fun n => F n a) Filter.atTop (𝓝 (f a))) : Filter.Tendsto (fun n => ∫ a, F n a ∂μ) Filter.atTop (𝓝 <| ∫ a, f a ∂μ)",
  "family": "Convergence theorems",
  "id": "library.mathlib.dominated-convergence-theorem.v001",
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  "landmark": {
    "completeFormalType": "theorem MeasureTheory.tendsto_integral_of_dominated_convergence {α G : Type*} [NormedAddCommGroup G] [NormedSpace ℝ G] {m : MeasurableSpace α} {μ : Measure α} {F : ℕ → α → G} {f : α → G} (bound : α → ℝ) (F_measurable : ∀ n, AEStronglyMeasurable (F n) μ) (bound_integrable : Integrable bound μ) (h_bound : ∀ n, ∀ᵐ a ∂μ, ‖F n a‖ ≤ bound a) (h_lim : ∀ᵐ a ∂μ, Filter.Tendsto (fun n => F n a) Filter.atTop (𝓝 (f a))) : Filter.Tendsto (fun n => ∫ a, F n a ∂μ) Filter.atTop (𝓝 <| ∫ a, f a ∂μ)",
    "concepts": [
      "almost-everywhere convergence",
      "dominating functions",
      "Bochner integration",
      "limit interchange",
      "strong measurability"
    ],
    "plainLanguage": "If a sequence of measurable vector-valued functions converges almost everywhere and every function stays beneath the same integrable envelope, then taking the limit and taking the integral commute.",
    "poster": {
      "alt": "An ivory analysis poster shows measurable functions beneath one integrable bound, almost-everywhere pointwise convergence, and the equality between the limit of integrals and the integral of the limit.",
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      "caption": "Almost-everywhere convergence plus one integrable dominator permits interchange of limit and Bochner integral.",
      "description": "The poster uses a fixed gold envelope around successive green function profiles, a small exceptional-set motif, and converging accumulation ribbons. The exact limit-interchange formula is central, while a footer excludes uniform-convergence, monotone-convergence, and rate claims.",
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      "generatedAt": "2026-07-25T05:35:05.555167910Z",
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      "sha256": "sha256:fae38bb1722e1803d12db070d2702009bab82b9b4a6a9062d5908aabda3066a9",
      "title": "Dominated Convergence Theorem at a glance",
      "width": 1024,
      "kind": "theorem_poster",
      "transcript": "MEASURE THEORY · LIMIT INTERCHANGE\nDOMINATED CONVERGENCE THEOREM\nFₙ strongly measurable almost everywhere\n‖Fₙ(a)‖ ≤ bound(a) almost everywhere\nbound integrable\nFₙ(a) → f(a) almost everywhere\nlimₙ ∫ Fₙ dμ = ∫ f dμ\nONE INTEGRABLE ENVELOPE CONTROLS THE LIMIT\nEXACT SCOPE\nSequential Bochner integrals. No convergence rate or uniform-convergence claim."
    },
    "visual": {
      "alt": "A sequence of varying function profiles remains inside one fixed integrable envelope and settles toward a limit while their accumulated areas converge.",
      "byteSize": 3009815,
      "caption": "One integrable envelope controls a pointwise-almost-everywhere limit strongly enough to pass the limit through the integral.",
      "description": "Several emerald function profiles evolve inside a fixed antique-gold upper envelope. A sparse muted set marks the almost-everywhere exception, while cobalt accumulation bands beneath the profiles converge to the limit profile's integral.",
      "derivatives": [
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      "generatedAt": "2026-07-25T05:22:45.478578719Z",
      "height": 1024,
      "mediaType": "image/png",
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      "publicPath": "/assets/library-theorems/dominated-convergence-theorem/theorem-schematic-v1.png",
      "sha256": "sha256:d9ead528c8db3067812b75130605b9c13ae53c9cd3666cc08bc1904885fa1181",
      "title": "Dominated convergence envelope schematic",
      "width": 1536
    },
    "walkthrough": null,
    "whyLandmark": "The dominated convergence theorem is one of measure theory's central limit-interchange principles. It turns pointwise information plus one global integrable bound into convergence of integrals and supports large parts of modern analysis and probability."
  },
  "nonClaims": [
    "Proof Atlas did not originate the dominated convergence theorem or Mathlib's declaration.",
    "The selected declaration requires its exact measurability, domination, integrability, and almost-everywhere convergence hypotheses.",
    "The generated explanation and visuals are not proof evidence."
  ],
  "publicExportEligible": true,
  "relationship": "upstream_direct",
  "schemaVersion": "library-theorem-entry-data.v1",
  "statementId": "statement.library.mathlib.dominated-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": "Almost-everywhere convergence under one integrable pointwise norm bound forces the Bochner integrals to converge to the integral of the limit.",
  "targetId": "target.library.mathlib.dominated-convergence-theorem.v001",
  "title": "Dominated Convergence Theorem",
  "upstreamOrigin": {
    "declarationName": "MeasureTheory.tendsto_integral_of_dominated_convergence",
    "immutableSourceUrl": "https://github.com/leanprover-community/mathlib4/blob/5e932f97dd25535344f80f9dd8da3aab83df0fe6/Mathlib/MeasureTheory/Integral/DominatedConvergence.lean#L58",
    "packageName": "mathlib",
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    "sourceFile": "Mathlib/MeasureTheory/Integral/DominatedConvergence.lean",
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}