Fubini and Tonelli: valid reduction of multiple Lebesgue integrals to iterated integrals

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.

Scientific 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

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.

Contributors

• The named authors in the research record · SCIENCE: 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.

• Sugli integrali multipli (RendLincei5,16I,608–614)
• Sull'integrazione per parti (RendLincei5,18II,246–253)

Provisional WRS assessment, edition 2026-10-23.4. Independent expert calibration is pending. This grade does not certify correctness or assign ownership.

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