Tonelli–Hobson Test

mathematics Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern mathematics ...

, the Tonelli–Hobson test gives sufficient criteria for a function ''ƒ'' on R² to be an integrable function. It is often used to establish that Fubini's theorem may be applied to ''ƒ''. It is named for

Leonida Tonelli Leonida Tonelli (19 April 1885 – 12 March 1946) was an Italian mathematician, noted for creating Tonelli's theorem, a variation of Fubini's theorem, and for introducing semicontinuity methods as a common tool for the direct method in the calc ...

and

E. W. Hobson Ernest William Hobson Fellow of the Royal Society, FRS (27 October 1856 – 19 April 1933) was an England, English mathematician, now remembered mostly for his books, some of which broke new ground in their coverage in English of topics fro ...

. More precisely, the Tonelli–Hobson test states that if ''ƒ'' is a real-valued

measurable function In mathematics and in particular measure theory, a measurable function is a function between the underlying sets of two measurable spaces that preserves the structure of the spaces: the preimage of any measurable set is measurable. This is in di ...

on R², and either of the two iterated integrals :

\int_\mathbb\left(\int_\mathbb, f(x,y), \,dx\right)\, dy

or :

\int_\mathbb\left(\int_\mathbb, f(x,y), \,dy\right)\, dx

is finite, then ''ƒ'' is

Lebesgue-integrable In mathematics, the integral of a non-negative function of a single variable can be regarded, in the simplest case, as the area between the graph of that function and the -axis. The Lebesgue integral, named after French mathematician Henri Leb ...

on R². Integral calculus Theorems in analysis {{mathanalysis-stub