Product measure

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Let and be two measurable spaces, that is, and are sigma algebras on and respectively, and let and be measures on these spaces. Denote by the sigma algebra on the Cartesian product generated by subsets of the form , where and This sigma algebra is called the tensor-product σ-algebra on the product space.

A product measure is defined to be a measure on the measurable space satisfying the property

for all


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

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.





  • 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


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