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 ${\displaystyle (X_{1},\Sigma _{1})}$ and ${\displaystyle (X_{2},\Sigma _{2})}$ be two measurable spaces, that is, ${\displaystyle \Sigma _{1}}$ and ${\displaystyle \Sigma _{2}}$ are sigma algebras on ${\displaystyle X_{1}}$ and ${\displaystyle X_{2}}$ respectively, and let ${\displaystyle \mu _{1}}$ and ${\displaystyle \mu _{2}}$ be measures on these spaces. Denote by ${\displaystyle \Sigma _{1}\otimes \Sigma _{2}}$ the sigma algebra on the Cartesian product ${\displaystyle X_{1}\times X_{2}}$ generated by subsets of the form ${\displaystyle B_{1}\times B_{2}}$, where ${\displaystyle B_{1}\in \Sigma _{1}}$ and ${\displaystyle B_{2}\in \Sigma _{2}.}$ This sigma algebra is called the tensor-product σ-algebra on the product space.

A product measure ${\displaystyle \mu _{1}\times \mu _{2}}$ (also denoted by ${\displaystyle \mu _{1}\otimes \mu _{2}}$ by many authors) is defined to be a measure on the measurable space ${\displaystyle (X_{1}\times X_{2},\Sigma _{1}\otimes \Sigma _{2})}$ satisfying the property

${\displaystyle (\mu _{1}\times \mu _{2})(B_{1}\times B_{2})=\mu _{1}(B_{1})\mu _{2}(B_{2})}$

for all

${\displaystyle B_{1}\in \Sigma _{1},\ B_{2}\in \Sigma _{2}}$.

(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 ${\displaystyle \sigma }$-finite, the product measure is uniquely defined, and for every measurable set E,

${\displaystyle (\mu _{1}\times \mu _{2})(E)=\int _{X_{2}}\mu _{1}(E^{y})\,d\mu _{2}(y)=\int _{X_{1}}\mu _{2}(E_{x})\,d\mu _{1}(x),}$

where ${\displaystyle E_{x}=\{y\in X_{2}|(x,y)\in E\}}$ and ${\displaystyle E^{y}=\{x\in X_{1}|(x,y)\in E\}}$, 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 ${\displaystyle (X_{1},\Sigma _{1},\mu _{1})}$ and ${\displaystyle (X_{2},\Sigma _{2},\mu _{2})}$ are σ-finite.

The Borel measures on the Euclidean space Rⁿ 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" a given measure into a family of measures that can be integrated to give the original measure.