Editing Fubini's theorem (section)

==For complete measures==
The versions of Fubini's and Tonelli's theorems above do not apply to integration on the product of the real line <math>\R</math> with itself with Lebesgue measure.  The problem is that Lebesgue measure on <math>\R\times\R</math> is not the product of Lebesgue measure on <math>\R</math> with itself, but rather the completion of this: a product of two complete measure spaces <math>X</math> and <math>Y</math> is not in general complete.  For this reason, one sometimes uses versions of Fubini's theorem for complete measures: roughly speaking, one replaces all measures with their completions.  The various versions of Fubini's theorem are similar to the versions above, with the following minor differences:
*Instead of taking a product <math>X\times Y</math> of two measure spaces, one takes the completion of the product.
*If <math>f</math> is measurable on the completion of <math>X\times Y</math> then its restrictions to vertical or horizontal lines may be non-measurable for a measure zero subset of lines, so one has to allow for the possibility that the vertical or horizontal integrals are undefined on a set of measure 0 because they involve integrating non-measurable functions. This makes little difference, because they can already be undefined due to the functions not being integrable. 
*One generally also assumes that the measures on <math>X</math> and <math>Y</math> are complete, otherwise the two partial integrals along vertical or horizontal lines may be well-defined but not measurable. For example, if <math>f</math> is the characteristic function of a product of a measurable set and a non-measurable set contained in a measure 0 set then its single integral is well defined everywhere but non-measurable.