Does Fubini's theorem require absolute integrability for both integration orders?

[Poopsilon]

So given $\int_c^d \int_a^b f(x,y)dxdy$, we can exchange the order of the integrals provided that $\int_c^d \int_a^b |f(x,y)|dxdy < \infty$. Does this less-than-infinity property have to hold for both orders of iteration i.e. for dxdy and dydx? Or can it be proven that if it's finite for one order than it's finite for the other?

 

[mathman]

So given $\int_c^d \int_a^b f(x,y)dxdy$, we can exchange the order of the integrals provided that $\int_c^d \int_a^b |f(x,y)|dxdy < \infty$. Does this less-than-infinity property have to hold for both orders of iteration i.e. for dxdy and dydx? Or can it be proven that if it's finite for one order than it's finite for the other?

Yes (for |f(x,y)|, the order doesn't matter).