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| In [[mathematics]], given two [[measurable space]]s and [[measure (mathematics)|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 [[Set (mathematics)|sets]] and the [[product topology]] of two topological spaces, except that there can be many natural choices for the product measure.
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| Let <math>(X_1, \Sigma_1)</math> and <math>(X_2, \Sigma_2)</math> be two [[measurable space]]s, that is, <math>\Sigma_1</math> and <math>\Sigma_2</math> are [[sigma algebra]]s on <math>X_1</math> and <math>X_2</math> respectively, and let <math>\mu_1</math> and <math>\mu_2</math> be measures on these spaces. Denote by <math>\Sigma_1 \otimes \Sigma_2</math> the sigma algebra on the [[Cartesian product]] <math>X_1 \times X_2</math> generated by [[subset]]s of the form <math>B_1 \times B_2</math>, where <math>B_1 \in \Sigma_1</math> and <math>B_2 \in \Sigma_2.</math> This sigma algebra is called the ''tensor-product σ-algebra'' on the product space.
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| A ''product measure'' <math>\mu_1 \times \mu_2</math> is defined to be a measure on the measurable space <math>(X_1 \times X_2, \Sigma_1 \otimes \Sigma_2)</math> satisfying the property
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| :<math> (\mu_1 \times \mu_2)(B_1 \times B_2) = \mu_1(B_1) \mu_2(B_2)</math>
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| for all
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| :<math> B_1 \in \Sigma_1,\ B_2 \in \Sigma_2. </math>
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| In fact, when the spaces are <math>\sigma</math>-finite, the product measure is uniquely defined, and for every measurable set ''E'',
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| :<math>(\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),</math>
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| where ''E''<sub>''x''</sub> = {''y''∈''X''<sub>2</sub>|(''x'',''y'')∈''E''}, and ''E''<sup>''y''</sup> = {''x''∈''X''<sub>1</sub>|(''x'',''y'')∈''E''}, which are both measurable sets.
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| 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]].
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| The [[Borel measure]] on the [[Euclidean space]] '''R'''<sup>''n''</sup> can be obtained as the product of ''n'' copies of the Borel measure on the [[real line]] '''R'''.
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| Even if the two factors of the product space are [[complete measure|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.
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| The opposite construction to the formation of the product of two measures is [[disintegration theorem|disintegration]], which in some sense "splits" a given measure into a family of measures that can be integrated to give the original measure.
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| ==Examples==
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| *Given two measure spaces, there is always a unique maximal product measure μ,sub>max</sub> on their product, with the property that if μ<sub>max</sub>(''A'') is finite for some measurable set ''A'', then μ<sub>max</sub>(''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 [[Caratheodory extension theorem]].
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| *There is always a unique minimal product measure μ<sub>min</sub>, given by μ<sub>min</sub>(''S'') = sup<sub>''A''⊂''S'', μ<sub>max</sub>(''A'') finite</sub> μ<sub>max</sub>(''A''), where ''A'' and ''S'' are assumed to be measurable.
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| *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.
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| ==See also==
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| * [[Fubini's theorem]]
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| == References ==
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| * {{cite book|last=Loève|first=Michel|authorlink=Michel Loève|title=Probability Theory vol. I |edition=4th |publisher=Springer |year=1977 |isbn=0-387-90210-4 |chapter=8.2. Product measures and iterated integrals |pages=135–137 |ref=loe1978}}
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| * {{cite book|last=Halmos|first=Paul|authorlink=Paul Halmos|title=Measure theory |publisher=Springer |year=1974 |isbn=0-387-90088-8 |chapter=35. Product measures |pages=143–145 |ref=loe1978}}
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| {{PlanetMath attribution|id=952|title=Product measure}}
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| [[Category:Measures (measure theory)]]
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| [[Category:Integral calculus]]
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Hello and welcome. My title is Figures Wunder. North Dakota is where me and my husband live. One of the extremely very best issues in the world for me is to do aerobics and now I'm attempting to make money with it. Bookkeeping is what I do.
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