# Tonelli–Hobson test

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In 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 and E. W. Hobson.

Mathematics includes the study of such topics as quantity, structure, space, and change.

In mathematical analysis Fubini's theorem, introduced by Guido Fubini in 1907, is a result that gives conditions under which it is possible to compute a double integral by using iterated integral. One may switch the order of integration if the double integral yields a finite answer when the integrand is replaced by its absolute value.

Leonida Tonelli 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 calculus of variations.

More precisely, the Tonelli–Hobson test states that if ƒ is a real-valued measurable function on R2, and either of the two iterated integrals

In mathematics, a real-valued function is a function whose values are real numbers. In other words, it is a function that assigns a real number to each member of its domain.

In mathematics and in particular measure theory, a measurable function is a function between two measurable spaces such that the preimage of any measurable set is measurable, analogously to the definition that a function between topological spaces is continuous if the preimage of each open set is open. In real analysis, measurable functions are used in the definition of the Lebesgue integral. In probability theory, a measurable function on a probability space is known as a random variable.

In calculus an iterated integral is the result of applying integrals to a function of more than one variable in a way that each of the integrals considers some of the variables as given constants. For example, the function , if is considered a given parameter, can be integrated with respect to , . The result is a function of and therefore its integral can be considered. If this is done, the result is the iterated integral

${\displaystyle \int _{\mathbb {R} }\left(\int _{\mathbb {R} }|f(x,y)|\,dx\right)\,dy}$

or

${\displaystyle \int _{\mathbb {R} }\left(\int _{\mathbb {R} }|f(x,y)|\,dy\right)\,dx}$

is finite, then ƒ is Lebesgue-integrable on R2.

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