Product measure

In mathematics, given two measurable spaces and measures on them, one can obtain the product measurable space and the product measure on that space. Conceptually, this is similar to defining the Cartesian product of sets and the product topology of two topological spaces.

Let $(X\_1,\; Sigma\_1)$ and $(X\_2,\; Sigma\_2)$ be two measurable spaces, that is, $Sigma\_1$ and $Sigma\_2$ are sigma algebras on $X\_1$ and $X\_2$ respectively, and let $mu\_1$ and $mu\_2$ be measures on these spaces. Denote by $Sigma\_1\; imes\; Sigma\_2$ the sigma algebra on the Cartesian product $X\_1\; imes\; X\_2$ generated by subsets of the form $B\_1\; imes\; B\_2$, where $B\_1\; in\; Sigma\_1$ and $B\_2\; in\; Sigma\_2.$

The "product measure" $mu\_1\; imes\; mu\_2$ is defined to be the unique measure on the measurable space $(X\_1\; imes\; X\_2,\; Sigma\_1\; imes\; Sigma\_2)$ satisfying the property

:$(mu\_1\; imes\; mu\_2)(B\_1\; imes\; B\_2)\; =\; mu\_1(B\_1)\; mu\_2(B\_2)$

for all

:$B\_1\; in\; Sigma\_1,\; B\_2\; in\; Sigma\_2.$

In fact, for every measurable set "E",

:$(mu\_1\; imes\; mu\_2)(E)\; =\; int\_\{X\_2\}\; mu\_1(E^y),mu\_2(dy)\; =\; int\_\{X\_1\}\; mu\_2(E\_\{x\}),mu\_1(dx),$

where "E"_"x" = {"y"∈"X"₂|("x","y")∈"E"}, and "E"^"y" = {"x"∈"X"₁|("x","y")∈"E"}, 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 case that both (X1,Σ1,μ1) and (X2,Σ2,μ2) are σ-finite.

The Borel measure on the Euclidean space R^"n" can be obtained as the product of "n" copies of the Borel measure 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" a given measure into a family of measures that can be integrated to give the original measure.

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