Editing Fubini's theorem (section)

==Product measures==
If <math>X</math> and ''<math>Y</math>'' are [[measure space]]s, there are several natural ways to define a [[product measure]] on the product <math>X\times Y</math>.

In [[Product (category theory)|the sense of category theory]], measurable sets in the product <math>X\times Y</math> of measure spaces are the elements of the [[σ-algebra]] generated by the products <math>A\times B</math>, where <math>A</math> is measurable in <math>X</math> and <math>B</math> is measurable in <math>Y</math>.

A measure ''μ'' on ''X''&nbsp;×&nbsp;''Y'' is called a '''product measure''' if ''μ''(''A''&nbsp;×&nbsp;''B'')&nbsp;=&nbsp;''μ''<sub>1</sub>(''A'')''μ''<sub>2</sub>(''B'') for measurable subsets ''A''&nbsp;⊂&nbsp;''X'' and ''B''&nbsp;⊂&nbsp;''Y'' and measures ''μ''<sub>1</sub> on ''X'' and ''μ''<sub>2</sub> on ''Y''. In general, there may be many different product measures on ''X''&nbsp;×&nbsp;''Y''. Fubini's theorem and Tonelli's theorem both require technical conditions to avoid this complication; the most common approach is to assume that all measure spaces are [[σ-finite measure|σ-finite]], in which case there is a unique product measure on ''X''×''Y''. There is always a unique maximal product measure on ''X''&nbsp;×&nbsp;''Y'', where the measure of a measurable set is the [[Infimum and supremum|inf]] of the measures of sets containing it that are countable unions of products of measurable sets. The maximal product measure can be constructed by applying [[Carathéodory's extension theorem]] to the additive function μ such that ''μ''(''A''&nbsp;×&nbsp;''B'')&nbsp;=&nbsp;''μ''<sub>1</sub>(''A'')''μ''<sub>2</sub>(''B'') on the ring of sets generated by products of measurable sets. (Carathéodory's extension theorem gives a measure on a measure space that in general contains more measurable sets than the measure space ''X''&nbsp;×&nbsp;''Y'', so strictly speaking, the measure should be restricted to the [[σ-algebra]] generated by the products ''A''&nbsp;×&nbsp;''B'' of measurable subsets of ''X'' and ''Y''.)

The product of two [[complete measure space]]s is not usually complete. For example, the product of the [[Lebesgue measure]] on the unit interval ''I'' with itself is not the Lebesgue measure on the square ''I''&nbsp;×&nbsp;''I''. There is a variation of Fubini's theorem for complete measures, which uses the completion of the product of measures rather than the uncompleted product.