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 product measure $\mu_1 \times \mu_2$ 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 $$ \mu_1 \times \mu_2(B_1 \times B_2) = \mu_1(B_1) \mu_2(B_2) \text{\ for all\ } B_1 \in \mathcal{B}_1(E_1),\ B_2 \in \mathcal{B}_2(E_2). $$