Product measure
In mathematics, given two measurable spaces and measures on them, one can obtain a product measurable space and a product measure on that space. Conceptually, this is similar to defining the Cartesian product of sets and the product topology of two topological spaces, except that there can be many natural choices for the product measure. Let formula_1 and formula_2 be two measurable spaces, that is, formula_3 and formula_4 are sigma algebras on formula_5 and formula_6 respectively, and let formula_7 and formula_8 be measures on these spaces. Denote by formula_9 the sigma algebra on the Cartesian product formula_10 generated  by subsets of the form formula_11, where formula_12 and formula_13 This sigma algebra is called the "tensor-product σ-algebra" on the product space. A "product measure" formula_14 is defined to be a measure on the measurable space formula_16 satisfying the property for all In fact, when the spaces are formula_19-finite, the product measure is uniquely defined, and for every measurable set "E", where formula_21 and formula_22, which are both measurable sets. The existence of this measure is guaranteed by the Hahn–Kolmogorov theorem. The uniqueness of product measure is guaranteed only in the case that both formula_23 and  formula_24 are σ-finite. The Borel measures on the Euclidean space R can be obtained as the product of "n" copies of Borel measures on the real line R. Even if the two factors of the product space are complete measure spaces, the product space may not be. Consequently, the completion procedure is needed to extend the Borel measure into the Lebesgue measure, or to extend the product of two Lebesgue measures to give the Lebesgue measure on the product space. The opposite construction to the formation of the product of two measures is disintegration, which in some sense "splits