Product measure

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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 (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 \otimes \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. This sigma algebra is called the tensor-product σ-algebra on the product space.

A product measure \mu_1 \times \mu_2 is defined to be a measure on the measurable space (X_1 \times X_2, \Sigma_1 \otimes \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 multiplying measures, some of which are infinite, we define the product to be zero if any factor is zero.)

In fact, when the spaces are \sigma-finite, the product measure is uniquely defined, and for every measurable set E,

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

where Ex = {yX2|(x,y)∈E}, and Ey = {xX1|(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 the 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.

Examples[edit]

  • Given two measure spaces, there is always a unique maximal product measure μmax on their product, with the property that if μmax(A) is finite for some measurable set A, then μmax(A) = μ(A) for any product measure μ. In particular its value on any measurable set is at least that of any other product measure. This is the measure produced by the Carathéodory extension theorem.
  • There is always a unique minimal product measure μmin, given by μmin(S) = supAS, μmax(A) finite μmax(A), where A and S are assumed to be measurable.
  • Here is an example where a product has more than one product measure. Take the product X×Y, where X is the unit interval with Lebesgue measure, and Y is the unit interval with counting measure and all sets measurable. Then for the minimal product measure the measure of a set is the sum of the measures of its horizontal sections, while for the maximal product measure a set has measure zero unless it is contained in the union of a countable number of horizontal lines and a set with projection onto X of measure 0. In particular the diagonal has measure 0 for the minimal product measure and measure infinity for the maximal product measure.

See also[edit]

References[edit]

This article incorporates material from Product measure on PlanetMath, which is licensed under the Creative Commons Attribution/Share-Alike License.