A measure μ on "X"×"Y" is called a product measure if μ("A"×"B")=μ1("A")μ2("B") for measurable subsets "A"⊂"X" and "B"⊂"Y" and measures µ1 on "X" and µ2 on "Y". In general there may be many different product measures on "X"×"Y". Fubini's theorem and Tonelli's theorem both need technical conditions to avoid this complication; the most common way is to assume all measure spaces are σ-finite, in which case there is a unique product measure on "X"×"Y". There is always a unique maximal product measure on "X"×"Y", where the measure of a measurable set is the 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"×"B")=μ1("A")μ2("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"×"Y", so strictly speaking the measure should be restricted to the σ-algebra generated by the products "A"×"B" of measurable subsets of "X" and "Y".)