Product measure

In mathematics, given two measurable spaces and 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 sets and the product topology of two topological spaces, except that there can be many natural choices for the product measure.

Let ${\displaystyle (X_{1},\Sigma _{1})}$ and ${\displaystyle (X_{2},\Sigma _{2})}$ be two measurable spaces, that is, ${\displaystyle \Sigma _{1}}$ and ${\displaystyle \Sigma _{2}}$ are sigma 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 ${\displaystyle \Sigma _{1}\otimes \Sigma _{2}}$ the sigma algebra on the Cartesian product ${\displaystyle X_{1}\times X_{2}}$ generated by subsets of the form ${\displaystyle B_{1}\times B_{2}}$, where ${\displaystyle B_{1}\in \Sigma _{1}}$ and ${\displaystyle B_{2}\in \Sigma _{2}.}$ This sigma algebra is called the tensor-product σ-algebra on the product space.

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