202409220109

Tags : Measure Theory

Product Measures

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Suppose we have two $\sigma$-finite measure spaces $\mathfrak{A}=⟨{\Omega }_1,\mathcal{A},{\mu }_1⟩$ and $\mathfrak{B}=⟨{\Omega }_2,\mathcal{B},{\mu }_2⟩$. We in some sense want to find the size of rectangles that can be formed in the product ${\Omega }_1\times {\Omega }_2$.

The set of all rectangles $A\times B$ where $A\in \mathcal{A}$ and $B\in \mathcal{B}$ is represented by the set $\mathcal{R}$. The $\sigma$-field for the set then becomes $\sigma (\mathcal{R})$ which is represented by $\mathcal{A}\otimes \mathcal{B}$.

Definition

Given two product measure $\mathfrak{A}$ and $\mathfrak{B}$, The product measure space is given by $⟨{\Omega }_1\times {\Omega }_2,\mathcal{A}\otimes \mathcal{B},{\mu }_1\otimes {\mu }_2⟩$

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References