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Mathematics · A guided explanation

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

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.

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Rows or columns?

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.

The condition does real work

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.

Why infinity changes the puzzle

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.

What Tonelli adds

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.

A small check with a clear limit

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.

Scientific scope

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.

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People and systems

G. Fubini; Leonida Tonelli

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