Editing Fubini's theorem (section)

===Failure of Tonelli's theorem for non σ-finite spaces===

Suppose that ''X'' is the unit interval with the Lebesgue measurable sets and Lebesgue measure, and ''Y'' is the unit interval with all the subsets measurable and the [[counting measure]], so that ''Y'' is not σ-finite. If ''f'' is the characteristic function of the diagonal of ''X''×''Y'', then integrating ''f'' along ''X'' gives the 0 function on ''Y'', but integrating ''f'' along ''Y'' gives the function 1 on ''X''. So, the two iterated integrals are different. This shows that Tonelli's theorem can fail for spaces that are not σ-finite no matter which product measure is chosen. The measures are both [[decomposable measure|decomposable]], showing that Tonelli's theorem fails for decomposable measures (which are slightly more general than σ-finite measures).