Fubini's theorem

Suppose you want the total weight of a rectangular sheet of varying density. You could add up the weights of vertical strips, or of horizontal strips — either way you get the same total. Fubini's theorem is this everyday fact made rigorous: a multiple integral can be computed by integrating one variable at a time, and the order of integration does not change the answer.

In the measure-theoretic form, if f is integrable over a product space (its absolute value has a finite double integral), then the double integral equals each of the two iterated integrals: integrate in y first then x, or x first then y, and you get the same number. The companion Tonelli's theorem allows the same exchange for non-negative measurable functions even before integrability is checked.

The integrability hypothesis is not optional. Without it the two iterated integrals can both exist yet disagree — a famous warning that you may not swap the order of integration freely. So the safe recipe is: confirm the integral of |f| is finite (often by Tonelli), then apply Fubini to compute, slicing whichever way is easier.