Tonelli–Hobson test explained

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.

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

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

or

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

is finite, then ƒ is Lebesgue-integrable on R².^[1]

Notes and References

1. Book: Poznyak . Alexander S. . Advanced Mathematical Tools for Control Engineers: Volume 1: Deterministic Systems . 7 July 2010 . Elsevier . 978-0-08-055610-9 . 352 . en.