# product measure

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 $\mu_{1}\times\mu_{2}$ 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

 $\mu_{1}\times\mu_{2}(B_{1}\times B_{2})=\mu_{1}(B_{1})\mu_{2}(B_{2})\text{\ % for all\ }B_{1}\in\mathcal{B}_{1}(E_{1}),\ B_{2}\in\mathcal{B}_{2}(E_{2}).$
Title product measure ProductMeasure 2013-03-22 12:00:33 2013-03-22 12:00:33 djao (24) djao (24) 7 djao (24) Definition msc 28A35