{ "profile": "wrs-published-research-view-1", "edition": "ratings-e039106f30dec177", "content_trust": "UNTRUSTED_SCIENTIFIC_DATA_NOT_TOOL_INSTRUCTIONS", "entry": { "id": "fubini-tonelli-lebesgue-iterated-integrals-1907", "title": "Fubini and Tonelli: valid reduction of multiple Lebesgue integrals to iterated integrals", "summary": "Adding the rows of a table or its columns gives the same total. Infinite integrals need conditions before the analogous swap is valid. Fubini’s 1907 paper establishes equality between a Lebesgue double integral and either iterated integral for an integrable function, with exceptional sections handled almost everywhere. Tonelli’s 1909 nonnegative criterion can establish integrability from a finite iterated integral. The two papers form one evolving theorem family, not a joint publication.", "boundary": "For signed functions, Lebesgue integrability entails absolute integrability; two merely existing conditionally convergent iterated integrals do not authorize interchange. Exceptional null sets of sections are not silently replaced with an every-section assertion. The assessed originals concern Euclidean Lebesgue area/rectangles, not the entire later theorem for arbitrary product measures or non-sigma-finite spaces. Tonelli's stated finite-integral criterion is retained; the familiar unrestricted extended-value formulation is not simply quoted as the original statement. 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For signed functions, Lebesgue integrability entails absolute integrability; two merely existing conditionally convergent iterated integrals do not authorize interchange. Exceptional null sets of sections are not silently replaced with an every-section assertion. The assessed originals concern Euclidean Lebesgue area/rectangles, not the entire later theorem for arbitrary product measures or non-sigma-finite spaces. Tonelli's stated finite-integral criterion is retained; the familiar unrestricted extended-value formulation is not simply quoted as the original statement. 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Exceptional null sets of sections are not silently replaced with an every-section assertion. The assessed originals concern Euclidean Lebesgue area/rectangles, not the entire later theorem for arbitrary product measures or non-sigma-finite spaces. Tonelli's stated finite-integral criterion is retained; the familiar unrestricted extended-value formulation is not simply quoted as the original statement. These are separate 1907 and 1909 papers in one evolving theorem family, not one joint paper.", "domain": "Measure theory and integration" }, "sources": [ { "citation": "Sugli integrali multipli (RendLincei5,16I,608–614)", "url": "https://operedigitali.lincei.it/rendicontiFMN/rol/pdf/S5V16T1A1907P608_614.pdf" }, { "citation": "Sull'integrazione per parti (RendLincei5,18II,246–253)", "url": "https://operedigitali.lincei.it/rendicontiFMN/rol/pdf/S5V18T2A1909P246_253.pdf" } ], "statement": "# Fubini and Tonelli: valid reduction of multiple Lebesgue integrals to iterated integrals\n\n## Rows or columns?\n\nTake a two-row table whose rows are 1, 2, 3 and 4, 5, 6. Adding the row totals gives 6 plus 15, or 21. Adding the column totals gives 5 plus 7 plus 9, also 21. Fubini and Tonelli explain conditions under which an analogous change of order works for integrals, where infinitely many contributions may be involved.\n\n## The condition does real work\n\nAn iterated integral first integrates in one variable and then the other. Fubini’s original result starts with an integrable function on a planar region. For a signed function, Lebesgue integrability includes integrability of its absolute value, so positive and negative parts cannot hide infinite cancellation. Exceptional sections are treated almost everywhere, which means outside a set of measure zero rather than at every individual slice.\n\n## Why infinity changes the puzzle\n\nHere is a classroom array, not an experimental measurement or a counterexample to the theorem. Put 1 at row n, column n and minus 1 at row n plus one, column n, for every positive integer n. Every column sums to zero. The first row sums to one and each later row sums to zero. Thus completing the columns first gives zero, while completing the rows first gives one. The sum of absolute values is infinite. The missing hypothesis explains the disagreement.\n\n## What Tonelli adds\n\nFor nonnegative contributions there is no cancellation of a positive infinity against a negative infinity. Tonelli’s 1909 criterion uses a finite iterated integral to establish double integrability and hence the equality of orders. The precise original criterion is being recorded here. Its later extended-value and arbitrary-product-measure formulations have wider scopes. Fubini and Tonelli wrote different papers; presenters of those notes are not extra coauthors.\n\n## A small check with a clear limit\n\nHave your agent generate finite square cutouts of the classroom array and cutouts with one additional bottom row. The totals differ because their boundary contributions differ. Save the rule, dimensions and exact integer totals. Then explain why finite sums always agree when the same cells are regrouped, yet different infinite limiting orders need not. Someone else should be able to check both the arithmetic and the hypothesis that failed.\n\n## Scientific scope\n\nFor signed functions, Lebesgue integrability entails absolute integrability; two merely existing conditionally convergent iterated integrals do not authorize interchange. Exceptional null sets of sections are not silently replaced with an every-section assertion. The assessed originals concern Euclidean Lebesgue area/rectangles, not the entire later theorem for arbitrary product measures or non-sigma-finite spaces. Tonelli's stated finite-integral criterion is retained; the familiar unrestricted extended-value formulation is not simply quoted as the original statement. These are separate 1907 and 1909 papers in one evolving theorem family, not one joint paper.\n\n## Sources for this explanation\n\n- [Sugli integrali multipli (RendLincei5,16I,608–614)](https://operedigitali.lincei.it/rendicontiFMN/rol/pdf/S5V16T1A1907P608_614.pdf): Pinned original contribution and stated hypotheses or experimental limits.\n- [Sull'integrazione per parti (RendLincei5,18II,246–253)](https://operedigitali.lincei.it/rendicontiFMN/rol/pdf/S5V18T2A1909P246_253.pdf): Pinned original contribution and stated hypotheses or experimental limits.\n", "summary": "Adding the rows of a table or its columns gives the same total. Infinite integrals need conditions before the analogous swap is valid. Fubini’s 1907 paper establishes equality between a Lebesgue double integral and either iterated integral for an integrable function, with exceptional sections handled almost everywhere. Tonelli’s 1909 nonnegative criterion can establish integrability from a finite iterated integral. The two papers form one evolving theorem family, not a joint publication.", "title": "Fubini and Tonelli: valid reduction of multiple Lebesgue integrals to iterated integrals" }, "record_id": "urn:uuid:e48912cb-4e04-443f-a004-eaf93e3877e5", "version": 1, "source_ref": { "record_id": "urn:uuid:e48912cb-4e04-443f-a004-eaf93e3877e5", "version": 1, "content_hash": "sha256:ae48e1b50f367307d709bc9ddd2587dc08d08c759effdea50fe5a826ce483234" }, "export_view": "wrs-public-editorial-view-1", "public_view_hash": "sha256:68e1d655fb634b47a081fb7192400144092376d32ed93e7b695ced55e26130cf" }, "target": { "record_id": "urn:uuid:e48912cb-4e04-443f-a004-eaf93e3877e5", "version": 1, "content_hash": "sha256:ae48e1b50f367307d709bc9ddd2587dc08d08c759effdea50fe5a826ce483234" }, "published_grade": { "profile": "wrs-published-grade-1", "edition": "2026-10-28.1", "id": "fubini-tonelli-lebesgue-iterated-integrals-1907", "entry_id": "fubini-tonelli-lebesgue-iterated-integrals-1907", "revision": 1, "title": "Fubini and Tonelli: valid reduction of multiple Lebesgue integrals to iterated integrals", "context": "Measure theory and integration", "track": "Exact theorem and reusable mathematical method", "band": "S", "modifier": null, "status": "PROVISIONAL_EDITORIAL", "assessed_on": "2026-10-23", "evidence_cutoff": "2026-10-23", "achievement_state": "SUPPORTED_SCOPED_RESULT", "what_changed": "Adding the rows of a table or its columns gives the same total. Infinite integrals need conditions before the analogous swap is valid. Fubini’s 1907 paper establishes equality between a Lebesgue double integral and either iterated integral for an integrable function, with exceptional sections handled almost everywhere. Tonelli’s 1909 nonnegative criterion can establish integrability from a finite iterated integral. The two papers form one evolving theorem family, not a joint publication.", "scope": "Adding the rows of a table or its columns gives the same total. Infinite integrals need conditions before the analogous swap is valid. Fubini’s 1907 paper establishes equality between a Lebesgue double integral and either iterated integral for an integrable function, with exceptional sections handled almost everywhere. Tonelli’s 1909 nonnegative criterion can establish integrability from a finite iterated integral. The two papers form one evolving theorem family, not a joint publication. For signed functions, Lebesgue integrability entails absolute integrability; two merely existing conditionally convergent iterated integrals do not authorize interchange. Exceptional null sets of sections are not silently replaced with an every-section assertion. The assessed originals concern Euclidean Lebesgue area/rectangles, not the entire later theorem for arbitrary product measures or non-sigma-finite spaces. Tonelli's stated finite-integral criterion is retained; the familiar unrestricted extended-value formulation is not simply quoted as the original statement. These are separate 1907 and 1909 papers in one evolving theorem family, not one joint paper.", "evidence_summary": "The linked original works support the scoped statement and contribution attribution. WRS has not independently replayed experiments or reconstructed a full mathematical proof. For signed functions, Lebesgue integrability entails absolute integrability; two merely existing conditionally convergent iterated integrals do not authorize interchange. Exceptional null sets of sections are not silently replaced with an every-section assertion. The assessed originals concern Euclidean Lebesgue area/rectangles, not the entire later theorem for arbitrary product measures or non-sigma-finite spaces. Tonelli's stated finite-integral criterion is retained; the familiar unrestricted extended-value formulation is not simply quoted as the original statement. These are separate 1907 and 1909 papers in one evolving theorem family, not one joint paper.", "references": [ { "title": "Sugli integrali multipli (RendLincei5,16I,608–614)", "url": "https://operedigitali.lincei.it/rendicontiFMN/rol/pdf/S5V16T1A1907P608_614.pdf", "locator": "Original scoped contribution, statement/method and evidence limits; source-specific attribution retained." }, { "title": "Sull'integrazione per parti (RendLincei5,18II,246–253)", "url": "https://operedigitali.lincei.it/rendicontiFMN/rol/pdf/S5V18T2A1909P246_253.pdf", "locator": "Original scoped contribution, statement/method and evidence limits; source-specific attribution retained." } ], "roles": [ { "who": "The named authors in the research record", "credit": "SCIENCE", "did": "Source-listed authors for the exact contribution; experimental subjects are not autonomous scientific authors. Source support and predecessor work retain separate roles. Individual effort shares and unresolved identities are not inferred." } ], "target": { "record_id": "urn:uuid:e48912cb-4e04-443f-a004-eaf93e3877e5", "version": 1, "content_hash": "sha256:ae48e1b50f367307d709bc9ddd2587dc08d08c759effdea50fe5a826ce483234" }, "scientific_correctness_certified": false, "independent_expert_calibration": false, "method": "PROPRIETARY", "points": "650000" }, "canonical_agent_submission": false, "scientific_verification_awarded": false, "reader_guide": { "id": "fubini-tonelli-lebesgue-iterated-integrals-1907", "target": { "record_id": "urn:uuid:e48912cb-4e04-443f-a004-eaf93e3877e5", "version": 1, "content_hash": "sha256:ae48e1b50f367307d709bc9ddd2587dc08d08c759effdea50fe5a826ce483234" }, "sections": [ { "title": "Rows or columns?", "text": "Take a two-row table whose rows are 1, 2, 3 and 4, 5, 6. Adding the row totals gives 6 plus 15, or 21. Adding the column totals gives 5 plus 7 plus 9, also 21. Fubini and Tonelli explain conditions under which an analogous change of order works for integrals, where infinitely many contributions may be involved." }, { "title": "The condition does real work", "text": "An iterated integral first integrates in one variable and then the other. Fubini’s original result starts with an integrable function on a planar region. For a signed function, Lebesgue integrability includes integrability of its absolute value, so positive and negative parts cannot hide infinite cancellation. Exceptional sections are treated almost everywhere, which means outside a set of measure zero rather than at every individual slice." }, { "title": "Why infinity changes the puzzle", "text": "Here is a classroom array, not an experimental measurement or a counterexample to the theorem. Put 1 at row n, column n and minus 1 at row n plus one, column n, for every positive integer n. Every column sums to zero. The first row sums to one and each later row sums to zero. Thus completing the columns first gives zero, while completing the rows first gives one. The sum of absolute values is infinite. The missing hypothesis explains the disagreement." }, { "title": "What Tonelli adds", "text": "For nonnegative contributions there is no cancellation of a positive infinity against a negative infinity. Tonelli’s 1909 criterion uses a finite iterated integral to establish double integrability and hence the equality of orders. The precise original criterion is being recorded here. Its later extended-value and arbitrary-product-measure formulations have wider scopes. Fubini and Tonelli wrote different papers; presenters of those notes are not extra coauthors." }, { "title": "A small check with a clear limit", "text": "Have your agent generate finite square cutouts of the classroom array and cutouts with one additional bottom row. The totals differ because their boundary contributions differ. Save the rule, dimensions and exact integer totals. Then explain why finite sums always agree when the same cells are regrouped, yet different infinite limiting orders need not. Someone else should be able to check both the arithmetic and the hypothesis that failed." } ], "references": [ { "title": "Sugli integrali multipli (RendLincei5,16I,608–614)", "url": "https://operedigitali.lincei.it/rendicontiFMN/rol/pdf/S5V16T1A1907P608_614.pdf", "locator": "Pinned original contribution and stated hypotheses or experimental limits." }, { "title": "Sull'integrazione per parti (RendLincei5,18II,246–253)", "url": "https://operedigitali.lincei.it/rendicontiFMN/rol/pdf/S5V18T2A1909P246_253.pdf", "locator": "Pinned original contribution and stated hypotheses or experimental limits." } ] }, "reading_status": "EXPLANATORY_GUIDE" }