Table of contents |

2 Counterexample 3 Explanation via Lebesgue theory 4 In the positive sense |

On the other hand, if we define

Perhaps surprisingly, in some cases yes, as an example shows:

Obviously the sign gets reversed if the order of iterated integration gets reversed, i.e., if "To give the analytic explanation: the double integral exists only if

and in that case, the double integral coincides in value with either of the two iterated integrals. Thus, whenever the two iterated integrals differ in value from each other, the double integral of the absolute value of the function must be infinite. See Fubini's theorem.
These, in which the roles of the two variables are uncoupled, present no problem in this context; and neither do their linear combinations. Quite generally, given compact spaces *X* and *Y*, we can use the Stone-Weierstrass theorem to show that such functions give a subalgebra of *C*(*X*×*Y*) that is dense in the uniform norm: or in other words any continuous function on *X*×*Y* can be uniformly approximated by sums of functions *f*(*x*)*g*(*y*).

This implies that double integrals behave rather well, at least on a large collection of 'test' functions.