Product measure

 ( Measures ) 

In mathematics, given two 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 $(X\_1,\; \backslash Sigma\_1)$ and $(X\_2,\; \backslash Sigma\_2)$ be two , that is, $\backslash Sigma\_1$ and $\backslash Sigma\_2$ are on $X\_1$ and $X\_2$ respectively, and let $\backslash mu\_1$ and $\backslash mu\_2$ be measures on these spaces. Denote by $\backslash Sigma\_1\; \backslash otimes\; \backslash Sigma\_2$ the sigma algebra on the Cartesian product $X\_1\; \backslash times\; X\_2$ generated by of the form $B\_1\; \backslash times\; B\_2$, where $B\_1\; \backslash in\; \backslash Sigma\_1$ and $B\_2\; \backslash in\; \backslash Sigma\_2$:

$\backslash Sigma\_1\; \backslash otimes\; \backslash Sigma\_2\; =\; \backslash sigma\backslash left(\; \backslash lbrace\; B\_1\; \backslash times\; B\_2\; \backslash mid\; B\_1\; \backslash in\; \backslash Sigma\_1,\; B\_2\; \backslash in\; \backslash Sigma\_2\; \backslash rbrace\; \backslash right)$

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

A product measure $\backslash mu\_1\; \backslash times\; \backslash mu\_2$ (also denoted by $\backslash mu\_1\; \backslash otimes\; \backslash mu\_2$ by many authors) is defined to be a measure on the measurable space $(X\_1\; \backslash times\; X\_2,\; \backslash Sigma\_1\; \backslash otimes\; \backslash Sigma\_2)$ satisfying the property

$(\backslash mu\_1\; \backslash times\; \backslash mu\_2)(B\_1\; \backslash times\; B\_2)\; =\; \backslash mu\_1(B\_1)\; \backslash mu\_2(B\_2)\; \backslash qquad\; (B\_1\; \backslash in\; \backslash Sigma\_1,\; B\_2\; \backslash in\; \backslash Sigma\_2)$.

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

$(\backslash mu\_1\; \backslash times\; \backslash mu\_2)(E)\; =\; \backslash int\_\{X\_2\}\; \backslash mu\_1(E^y)\backslash ,d\backslash mu\_2(y)\; =\; \backslash int\_\{X\_1\}\; \backslash mu\_2(E\_\{x\})\backslash ,d\backslash mu\_1(x),$

where $E\_x\; =\; \backslash \{y\; \backslash in\; X\_2\; |\; (x,\; y)\; \backslash in\; E\backslash \}$ and $E^y\; =\; \backslash \{x\; \backslash in\; X\_1\; |\; (x,\; y)\; \backslash in\; E\backslash \}$, 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 $(X\_1,\; \backslash Sigma\_1,\; \backslash mu\_1)$ and $(X\_2,\; \backslash Sigma\_2,\; \backslash mu\_2)$ are σ-finite.

The on the Euclidean space R ⁿ 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, 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.

• Given two measure spaces, there is always a unique maximal product measure μ_max on their product, with the property that if μ_max( A) is finite for some measurable set A, then μ_max( 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.
• Sometimes there is also a unique minimal product measure μ_min, given by μ_min( S) = sup _{A⊂ S, μmax( A) finite} μ_max( 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 are 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 infinity unless it is contained in the union of a countable number of sets of the form A× B, where either A has Lebesgue measure 0 or B is a single point. (In this case the measure may be finite or infinite.) In particular, the diagonal has measure 0 for the minimal product measure and measure infinity for the maximal product measure.

• Fubini's theorem

• Michel Loève (1977). 9780387902104, Springer. ISBN 9780387902104
• Paul Halmos (1974). 9780387900889, Springer. . ISBN 9780387900889