Product measure

From Wikipedia, the free encyclopedia

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 (X1,Σ1) and (X2,Σ2) be two measure spaces, that is, Σ1 and Σ2 are sigma algebras on X1 and X2 respectively, and let μ1 and μ2 be measures on these spaces. Denote by \Sigma_1 \times \Sigma_2 the sigma algebra on the Cartesian product X_1 \times X_2 generated by subsets of the form B_1 \times B_2, where B_1 \in \Sigma_1 and B_2 \in \Sigma_2.

The product measure \mu_1 \times \mu_2 is defined to be the unique measure on the measurable space (X_1 \times X_2, \Sigma_1 \times \Sigma_2) satisfying the property

(\mu_1 \times \mu_2)(B_1 \times 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 \times \mu_2)(E) = \int_{X_2} \mu_1(E^y)\,\mu_2(dy) = \int_{X_1} \mu_2(E_{x})\,\mu_1(dx),

where Ex = {y∈X2|(x,y)∈E}, and Ey = {x∈X1|(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 Rn 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.

This article incorporates material from Product measure on PlanetMath, which is licensed under the GFDL.