Tonelli–Hobson test

In mathematics, the Tonelli–Hobson test gives sufficient criteria for a function &fnof; on R2 to be an integrable function. It is often used to establish that Fubini's theorem may be applied to &fnof;. It is named for Leonida Tonelli and E. W. Hobson.

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


 * $$\int_\mathbb{R}\left(\int_\mathbb{R}|f(x,y)|\,dx\right)\, dy$$

or


 * $$\int_\mathbb{R}\left(\int_\mathbb{R}|f(x,y)|\,dy\right)\, dx$$

is finite, then &fnof; is Lebesgue-integrable on R2.