product measure

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

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

$${\mu }_1\times {\mu }_2(B_1\times B_2)={\mu }_1(B_1){\mu }_2(B_2)\text{ for all }{B}_1\in {ℬ}_1(E_1),B_2\in {ℬ}_2(E_2).$$