# When did Fubini's name get applied to the theorem without measures?

Fubini's theorem, from 1907, expresses integration with respect to a product measure in terms of iterated integrals. The simpler version of this theorem for multiple Riemann integrals was used long before Fubini was around and of course was not known by his name. Nowadays it is common for the relation between multiple Riemann integrals and iterated integrals to be called Fubini's theorem in books.

A colleague of mine asked me when the label "Fubini's theorem" was first applied to this theorem about multiple Riemann integrals. (He considers it something of a travesty to use Fubini's name for this result in multivariable calculus books, where there is no measure theory content. As an example, in the 4th edition of Calculus (1990) by Larson, Hostetler, and Edwards the authors write "The following theorem was proved by the Italian mathematician Guido Fubini" and then they give a theorem on double integrals of continuous functions which certainly was not proved by Fubini.) I found this theorem does not have Fubini's name in some calculus and analysis books written decades ago: Whittaker and Watson's Modern Analysis (4th ed., 1927), Volume II of Apostol's Calculus (1962), Rudin's Principle of Mathematical Analysis (3rd ed., 1964), Thomas's Calculus and Analytic Geometry (4th ed., 1969), Bers's Calculus (1969), Loomis's Calculus (1974), Sherman Stein's Calculus and Analytic Geometry (2nd ed., 1977), George Simmons's Calculus with Analytic Geometry (1985), Marsden and Weinstein's Calculus III (1985), and Leithold's The Calculus with Analytic Geometry (5th ed., 1986). They all call this result something like "the theorem on iterated integrals".

I found the name "Fubini's theorem" used for multiple Riemann integrals in Spivak's Calculus on Manifolds (1965). Does anyone know of an earlier usage of the label "Fubini's theorem" for multiple Riemann integrals?

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I want to clarify that the OP is interested in references that use the name "Fubini's theorem" for multiple Riemann integrals. He is not interested in *why* the switch happened, or how the label is justified. –  Srivatsan Nov 7 '11 at 21:45
That's true. The justification for the label is obvious, anyway. –  KCd Nov 8 '11 at 3:46

There is a note by R. T. Seeley in the American Mathematical Monthly from 1961 (vol. 68, pp. 56-57) titled Fubini Implies Leibniz Implies $F_{yx}=F_{xy}$. The note can be found in Selected Papers on Calculus (1969), edited by a committee chaired by Tom Apostol. I quote from the second paragraph:

"The note first assumes a simple form of Fubini's theorem for Riemann integrals (A), uses this to prove Leibniz's rule (B), and uses this in turn to prove one of the stronger forms of the theorem on mixed partials (C). It then states the corresponding results for Lebesgue integrals; the proofs remain the same."

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Thank you for this earlier data point. How did you find this, or did you know of it independently (before reading my question)? –  KCd Jan 8 '13 at 20:22
I had the book with selected papers from Monthly and Mathematics Magazine in my bookshelf. It's organized thematically, so I only had to look up the section on multiple integration. –  Per Manne Jan 8 '13 at 20:59

I used a Google search on 'Fubini 's Theorem' on books from 1947 to 1976 and read the snippets of text that appear. This is the only text before 1965 that intrigued me.

The author says 'We establish first a basic theorem, whose title is borrowed from a similar but more elegant theorem in Lebesgue Theory, proved by the Italian mathematician G. Fubini in 1910'.

Checking the library for the book lead me to a free online version. Here is the page from the Google search snippet.

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Here's what I can verify: Fubini apparently proved the general form of the theorem in 1907 and began dispersing the proof in oral lectures. It did not appear in a fully detailed statement and proof in the literature until 1958. It is sometimes called the Tonnelli theorem because Fubini's result is actually a modified form of the result first proved on product measure spaces by Tonnelli. I cannot trace when the name of the theorem first appears in a calculus text. i do know that in the 8th edition of the famous calculus book by George B Thomas and Finney, it appears,but apparently this was a recent edition by Finney and it was already common in calculus texts by then. I can be certain that it appeared by name in Spivak's Calculus on Manifolds (it also appears there on page 59 of the Westview edition; where Spivak says the theorem is "a special case of a theorem proved by Fubini long after [the theorem] was known." (Westview Press edition, page 57-58). It was also called this in James Munkres' Analysis On Manifolds and the earlier Functions Of Several Variables by Wendell Fleming. Fubini published a detailed proof in the modern language of product measures-building on Tonnelli's improvements-in Fubini, G. "Sugli integrali multipli." Opere scelte, Vol. 2. Cremonese, pp. 243-249, 1958. This seems to be the root point where it became standard practice to call the multiple integral theorem Fubini's theorem. There is oral tradition that in honors advanced calculus lectures at Princeton and Harvard in the early 1960's, it was referred to as Fubini's theorem as well, but I cannot verify this. Also, Loomis and Sternberg do not call it this in their famous text, which is based on those lectures at Harvard. This tradition is usually attributed to Spivak in his remembrances as a graduate student at Princeton, but I cannot verify this.

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I don't know about the "After that, it became standard practice" bit, but the earlier part intrigues me. On what basis do you assert that the Thomas calculus book is the first calculus book which called the result Fubini's theorem? –  KCd Nov 8 '11 at 3:45
On what basis do you claim there is an "oral tradition"? You've made a number of historical claims in the past that you've "heard somewhere" that turned out to be false. Please cite some source (similarly for the claim about Thomas's books, though KCd beat me to it). –  Adam Smith Nov 8 '11 at 3:52
It seems a bit unusual for you to write that you can be sure the term was used in Spivak's Calculus on Manifolds since I already said as much in my question. A web search says Tonelli gave the first correct proof in 1909, so saying that what Fubini did is a modified version of what Tonelli did, at the start of your answer, seems backwards. And 1958 is just the year that Fubini's old 7-page paper from 1907 appeared in his Selected Works (Opere scelte), so 1958 -- 50 years later -- is not the first year a correct proof was in print (which would surprise Tonelli, perhaps). Fubini died in 1943. –  KCd Nov 8 '11 at 12:48
How on earth does this answer have 4 upvotes?!?!?!? –  Adam Smith Nov 8 '11 at 16:04
I'm just asking for the earliest reference people know where Fubini's name is attached to Riemann integrals. I started off with Spivak's name as one data point directly in my question, from his book in 1965. I'm interested in any earlier data points that people know. Books from the 1970s, 80s, and 90s are not relevant to what I am seeking. –  KCd Nov 9 '11 at 0:57