{
  "artifactId": "artifact.library.mathlib.fubini-theorem.v001",
  "candidateOnly": false,
  "category": "Measure theory and analysis",
  "claimBoundary": "This page indexes Mathlib's Fubini theorem for a Bochner-integrable real-normed-space-valued function on the product of two s-finite measure spaces. The selected equality integrates first in β with ν and then in α with μ. The declaration does not assume completeness; its substantive Fubini content is the complete-valued case, while Mathlib's Bochner integral is definitionally zero for incomplete codomains. It does not state Tonelli's theorem for nonnegative functions, remove integrability, or include the symmetric order in the same declaration.",
  "collectionSlug": "landmark-theorems-in-mathlib",
  "exactFormalStatement": "theorem MeasureTheory.integral_prod {α β E : Type*} [MeasurableSpace α] [MeasurableSpace β] {μ : Measure α} {ν : Measure β} [NormedAddCommGroup E] [NormedSpace ℝ E] [SFinite ν] [SFinite μ] (f : α × β → E) (hf : Integrable f (μ.prod ν)) : ∫ z, f z ∂μ.prod ν = ∫ x, ∫ y, f (x, y) ∂ν ∂μ",
  "family": "Product integration",
  "id": "library.mathlib.fubini-theorem.v001",
  "links": {
    "collection": "/collections/landmark-theorems-in-mathlib/",
    "collectionData": "/data/collections/mathlib-landmarks.json",
    "entry": "/library-theorems/fubini-theorem/",
    "entryData": "/data/library-theorems/fubini-theorem.json",
    "evidence": "/proofs/artifact.library.mathlib.fubini-theorem.v001.html",
    "evidenceData": "/data/proofs/artifact.library.mathlib.fubini-theorem.v001.evidence.json",
    "source": "/sources/upstream/fubini-theorem/"
  },
  "landmark": {
    "completeFormalType": "theorem MeasureTheory.integral_prod {α β E : Type*} [MeasurableSpace α] [MeasurableSpace β] {μ : Measure α} {ν : Measure β} [NormedAddCommGroup E] [NormedSpace ℝ E] [SFinite ν] [SFinite μ] (f : α × β → E) (hf : Integrable f (μ.prod ν)) : ∫ z, f z ∂μ.prod ν = ∫ x, ∫ y, f (x, y) ∂ν ∂μ",
    "concepts": [
      "product measures",
      "iterated integrals",
      "Bochner integration",
      "s-finite measures",
      "integrability"
    ],
    "plainLanguage": "When a function on a product space is integrable, its total integral can be computed by first integrating along one coordinate and then integrating the result along the other.",
    "poster": {
      "alt": "An ivory measure-theory poster compares one product-measure integral of f on α times β with an iterated integral first over β and then over α.",
      "byteSize": 2898575,
      "caption": "Product integration equals the selected iterated integration order for an integrable function on s-finite measure spaces.",
      "description": "The poster treats the product surface and its family of coordinate slices as two views of the same accumulation. The exact equality is central, the order ν then μ is visually explicit, and the footer distinguishes this integrable Bochner form from Tonelli and the symmetric companion.",
      "derivatives": [
        {
          "byteSize": 112940,
          "height": 960,
          "mediaType": "image/webp",
          "publicPath": "/assets/library-theorems/fubini-theorem/theorem-poster-v1-640.webp",
          "sha256": "sha256:e5572f1a93221b4fad68a7edffc75cf94c87f286b873eba42826c4feda09c66f",
          "width": 640
        },
        {
          "byteSize": 276568,
          "height": 1536,
          "mediaType": "image/webp",
          "publicPath": "/assets/library-theorems/fubini-theorem/theorem-poster-v1-1024.webp",
          "sha256": "sha256:43dc78ba9142f8d0708e37af7c094fbd9c6e3cd8f9bc72388ba6b107e900065c",
          "width": 1024
        }
      ],
      "generatedAt": "2026-07-25T05:41:14.548400213Z",
      "height": 1536,
      "mediaType": "image/png",
      "modelRequested": "gpt-image-2",
      "publicPath": "/assets/library-theorems/fubini-theorem/theorem-poster-v1.png",
      "sha256": "sha256:c9b3506605ce8aa7a4f564a30bd34d535dc4587321610affc54286398057f9ff",
      "title": "Fubini's Theorem at a glance",
      "width": 1024,
      "kind": "theorem_poster",
      "transcript": "MEASURE THEORY · PRODUCT INTEGRATION\nFUBINI'S THEOREM\nf : α × β → E\nμ and ν s-finite\nf integrable for μ × ν\n∫ f d(μ × ν) = ∫ₓ (∫ᵧ f(x,y) dν) dμ\nONE PRODUCT INTEGRAL · ONE ITERATED INTEGRAL\nORDER IN THIS DECLARATION\nFirst β with ν · then α with μ\nEXACT SCOPE\nIntegrable Bochner form. Tonelli and the symmetric order are separate statements."
    },
    "visual": {
      "alt": "A continuous surface over a product grid is accumulated first along one family of slices and then across the remaining coordinate, matching one total product-space volume.",
      "byteSize": 2553002,
      "caption": "For an integrable function, product-space accumulation equals accumulation by successive coordinate slices.",
      "description": "A smooth emerald-and-cobalt surface rises over an ivory product grid. Parallel slices are gathered into one intermediate ribbon, and those ribbons are then accumulated into a single antique-gold total matching the full product-space integral.",
      "derivatives": [
        {
          "byteSize": 41212,
          "height": 427,
          "mediaType": "image/webp",
          "publicPath": "/assets/library-theorems/fubini-theorem/theorem-schematic-v1-640.webp",
          "sha256": "sha256:098883959bd2114393165be022e0d488618a40d746d5ceb0646f7146db7a47a4",
          "width": 640
        },
        {
          "byteSize": 130544,
          "height": 800,
          "mediaType": "image/webp",
          "publicPath": "/assets/library-theorems/fubini-theorem/theorem-schematic-v1-1200.webp",
          "sha256": "sha256:db25a2d46ec5e547e64ac64b2e519e9f57612142a869bebcecf6938e45a904be",
          "width": 1200
        }
      ],
      "generatedAt": "2026-07-25T05:27:57.761351324Z",
      "height": 1024,
      "mediaType": "image/png",
      "modelRequested": "gpt-image-2",
      "publicPath": "/assets/library-theorems/fubini-theorem/theorem-schematic-v1.png",
      "sha256": "sha256:396c97ecdb3ad219bfaa303d9a9c77b8eb43dc4379953415fb6fc5a8ea7c863f",
      "title": "Product-to-iterated integration schematic",
      "width": 1536
    },
    "walkthrough": null,
    "whyLandmark": "Fubini's theorem is the central bridge between integration on product spaces and repeated one-variable integration. It supports multivariable analysis, probability, partial differential equations, and almost every subject that uses product measures."
  },
  "nonClaims": [
    "Proof Atlas did not originate Fubini's theorem or Mathlib's declaration.",
    "The selected declaration requires integrability on the product measure and records one order of iteration; the symmetric order is a companion theorem.",
    "The generated explanation and visuals are not proof evidence."
  ],
  "publicExportEligible": true,
  "relationship": "upstream_direct",
  "schemaVersion": "library-theorem-entry-data.v1",
  "statementId": "statement.library.mathlib.fubini-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": "For an integrable real-normed-space-valued function on a product of two s-finite measure spaces, the product-measure Bochner integral equals the iterated integral in the selected order.",
  "targetId": "target.library.mathlib.fubini-theorem.v001",
  "title": "Fubini's Theorem",
  "upstreamOrigin": {
    "declarationName": "MeasureTheory.integral_prod",
    "immutableSourceUrl": "https://github.com/leanprover-community/mathlib4/blob/5e932f97dd25535344f80f9dd8da3aab83df0fe6/Mathlib/MeasureTheory/Integral/Prod.lean#L494",
    "packageName": "mathlib",
    "repositoryUrl": "https://github.com/leanprover-community/mathlib4",
    "sourceArtifactHash": "sha256:3f695c14e45e3e97e28df9e90bd6db4d0283ced3db5572c67ca67f4297f0e1f9",
    "sourceByteLength": 33754,
    "sourceCommit": "5e932f97dd25535344f80f9dd8da3aab83df0fe6",
    "sourceFile": "Mathlib/MeasureTheory/Integral/Prod.lean",
    "sourceLine": 494,
    "verificationKind": "git_worktree"
  },
  "publicPresentationReview": {
    "status": "reviewed"
  }
}