PlanetMath: product measure

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product measure

(Definition)

Let $(E_1, \mathcal{B}_1(E_1))$ and $(E_2, \mathcal{B}_2(E_2))$ be two measurable spaces, with measures $\mu_1$ and $\mu_2$ . Let $\mathcal{B}_1 \times \mathcal{B}_2$ be the sigma algebra on $E_1 \times E_2$ generated by subsets of the form $B_1 \times B_2$ , where $B_1 \in \mathcal{B}_1(E_1)$ and $B_2 \in \mathcal{B}_2(E_2)$ .

The product measure $\mu_1 \times \mu_2$ is defined to be the unique measure on the measurable space $(E_1 \times E_2, \mathcal{B}_1 \times \mathcal{B}_2)$ satisfying the property

$\displaystyle \mu_1 \times \mu_2(B_1 \times B_2) = \mu_1(B_1) \mu_2(B_2)$ for all $\displaystyle B_1 \in \mathcal{B}_1(E_1),\ B_2 \in \mathcal{B}_2(E_2).$

"product measure" is owned by djao.

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Attachments:: infinite product measure (Definition) by CWoo

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Cross-references: property, subsets, generated by, sigma algebra, measures, measurable spaces
There are 6 references to this entry.

This is version 3 of product measure, born on 2001-11-17, modified 2004-04-05.
Object id is 952, canonical name is ProductMeasure.
Accessed 5431 times total.

Classification:

AMS MSC:

28A35 (Measure and integration :: Classical measure theory :: Measures and integrals in product spaces)

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