Product measure

In mathematics, given two measurable spaces and measures on them, one can obtain the product measurable space and the product measure on that space. Conceptually, this is similar to defining the Cartesian product of sets and the product topology of two topological spaces.

Let $(X_1, \Sigma_1)$ and $(X_2, \Sigma_2)$ be two measure spaces, that is, $\Sigma_1$ and $\Sigma_2$ are sigma algebras on $X_1$ and $X_2$ respectively, and let $\mu_1$ and $\mu_2$ be measures on these spaces. Denote by $\Sigma_1 \times \Sigma_2$ the sigma algebra on the Cartesian product $X_1 \times X_2$ generated by subsets of the form $B_1 \times B_2$ , where $B_1 \in \Sigma_1$ and $B_2 \in \Sigma_2.$

The product measure $\mu_1 \times \mu_2$ is defined to be the unique measure on the measurable space $(X_1 \times X_2, \Sigma_1 \times \Sigma_2)$ satisfying the property

(\mu_1 \times \mu_2)(B_1 \times B_2) = \mu_1(B_1) \mu_2(B_2)

for all

B_1 \in \Sigma_1,\ B_2 \in \Sigma_2.

In fact, for every measurable set E,

(\mu_1 \times \mu_2)(E) = \int_{X_2} \mu_1(E^y)\,\mu_2(dy) = \int_{X_1} \mu_2(E_{x})\,\mu_1(dx),

where E_x = {y∈X₂|(x,y)∈E}, and E^y = {x∈X₁|(x,y)∈E}, which are both measurable sets.

The existence and uniqueness of this measure is guaranteed by the Hahn-Kolmogorov theorem.

The Borel measure on the Euclidean space Rⁿ can be obtained as the product of n copies of the Borel measure on the real line R.

Measure theory | Integral calculus

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