Talk:Tonelli–Hobson test

This page is in need of considerable revision for the following reasons:

(1) In both exposed expressions a right vertical bar (indicating together with the left vertical bar, the absolute value of function $$\ f(x,y)$$) is to be inserted.

(2) Rather than $$\ R^2$$, one should consider any measurable set, say $$\ E$$.

(3) I believe that the attribution of this test to Tonelli is incorrect: the correct attibution is Hobson test or Hobson theorem. See


 * E. W. Hobson, Proc. Lond. Math. Soc. (2), Vol. VIII (1909), p. 22.

See also:


 * E. W. Hobson, The theory of functions of a real variable and the theory of Fourier's series, Third Edition, Vol. I (Cambridge University Press, 1950), section 429 (pp. 629-632).

I believe that the confusion may have arisen owing to two reasons: firstly, Fubini has a similar, but distinct, theorem as Hobson (Rend. del. Real. Accad. dei Lincei (5), Vol. XVI (1910), p. 608) [note the dates 1909 and 1910!] to which indeed Hobson refers in the above-cited book, and secondly, Fubini and Tonelli have a theorem (Rend. del Circ. Mat. di Palermo, Vol. XL (1916), p. 295) related to primitive functions, or indefinite integrals, when integration is in the sense of Lebesgue. For the latter see Section 418 (pp. 608-614) of the above-cited book by Hobson. It appears that the association of Tonelli with Fubini, through the last-mentioned theorem, may have led to, firstly, confusing Tonelli with Fubini, and, secondly, confusing a theorem due to Hobson with a related theorem by Fubini and thus mistakenly (for two reasons) arriving at the designation Tonelli-Hobson test. At least Hobson-Tonelli test would be less unfair.

Since I would drastically change this article, I have decided to leave the task of revision to those who have substantially contributed to it. Three different pages may be appropriate, in each dealing with: (i) the Hobson theorem, (ii) the Fubini theorem and (iii) the Fubini-Tonelli theorem.

--BF 00:36, 6 December 2006 (UTC)