The method of exausted in double integrals

We've learned about improper double integrals and I've a couple of questions :-)

If a function is non-negative or non-positive then if we integrate it over an unbounded domain, the outcome won't depend on what type of exaustion we chose.. do you know what theorem is that ?

I've tried to think a function that didn't satisfy that condition and its integration depends on the type of exaustion but I could think of one (is there a method for doing this?)

thank you in advance