Editing Fubini's theorem (section)

===Failure of Tonelli's theorem for non-measurable functions===

Suppose that ''X'' is the first uncountable ordinal, with the finite measure where the measurable sets are either countable (with measure 0) or the sets of countable complement (with measure 1). The (non-measurable) subset ''E'' of ''X''×''X'' given by pairs (''x'' , ''y'') with ''x''<''y'' is countable on every horizontal line and has countable complement on every vertical line. If ''f'' is the characteristic function of ''E'' then the two iterated integrals of ''f'' are defined and have different values 1 and 0. The function ''f'' is not measurable. This shows that Tonelli's theorem can fail for non-measurable functions.