Inmathematics, given twomeasurable spaces andmeasures on them, one can obtain aproduct measurable space and aproduct measure on that space. Conceptually, this is similar to defining theCartesian product ofsets and theproduct topology of two topological spaces, except that there can be many natural choices for the product measure.

Let${\displaystyle (X_1,{\mathrm{\Sigma }}_1)}$ and${\displaystyle (X_2,{\mathrm{\Sigma }}_2)}$ be twomeasurable spaces, that is,${\displaystyle {\mathrm{\Sigma }}_1}$ and${\displaystyle {\mathrm{\Sigma }}_2}$ aresigma 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${\displaystyle {\mathrm{\Sigma }}_1\otimes {\mathrm{\Sigma }}_2}$ the sigma algebra on theCartesian product${\displaystyle X_1\times X_2}$ generated bysubsets of the form${\displaystyle B_1\times B_2}$, where${\displaystyle B_1\in {\mathrm{\Sigma }}_1}$ and${\displaystyle B_2\in {\mathrm{\Sigma }}_2}$:

${\displaystyle {\mathrm{\Sigma }}_1\otimes {\mathrm{\Sigma }}_2=\sigma \left(\{B_1\times B_2\mid B_1\in {\mathrm{\Sigma }}_1,B_2\in {\mathrm{\Sigma }}_2\}\right)}$

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

Aproduct 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${\displaystyle (X_1\times X_2,{\mathrm{\Sigma }}_1\otimes {\mathrm{\Sigma }}_2)}$ satisfying the property

${\displaystyle ({\mu }_1\times {\mu }_2)(B_1\times B_2)={\mu }_1(B_1){\mu }_2(B_2)(B_1\in {\mathrm{\Sigma }}_1,B_2\in {\mathrm{\Sigma }}_2)}$.

(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 setE,

${\displaystyle ({\mu }_1\times {\mu }_2)(E)={\int }_{X_2}{\mu }_1(E^y)d{\mu }_2(y)={\int }_{X_1}{\mu }_2(E_x)d{\mu }_1(x),}$

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 theHahn–Kolmogorov theorem. The uniqueness of product measure is guaranteed only in the case that both${\displaystyle (X_1,{\mathrm{\Sigma }}_1,{\mu }_1)}$ and${\displaystyle (X_2,{\mathrm{\Sigma }}_2,{\mu }_2)}$ areσ-finite.

TheBorel measures on theEuclidean spaceRⁿ can be obtained as the product ofn copies of Borel measures on thereal lineR.

Even if the two factors of the product space arecomplete measure spaces, the product space may not be. Consequently, the completion procedure is needed to extend the Borel measure into theLebesgue 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 isdisintegration, 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 setA, 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 theCarathéodory 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), whereA andS are assumed to be measurable.
• Here is an example where a product has more than one product measure. Take the productX×Y, whereX is the unit interval with Lebesgue measure, andY 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 formA×B, where eitherA has Lebesgue measure 0 orB 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

• Loève, Michel (1977). "8.2. Product measures and iterated integrals".Probability Theory vol. I (4th ed.). Springer. pp. 135–137.ISBN 0-387-90210-4.
• Halmos, Paul (1974). "35. Product measures".Measure theory. Springer. pp. 143–145.ISBN 0-387-90088-8.

This article incorporates material from Product measure onPlanetMath, which is licensed under theCreative Commons Attribution/Share-Alike License.

• Measures (measure theory)
• Integral calculus

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