I got stuck on this problem. For the first statement, I tried to use $\epsilon -\delta$ condition, but still couldn't come to conclusion. So can anyone please help me solve this or give me some clue how to solve this. Thanks so much. I really appreciate.
For $j = 1,2$, let $\mu_j, v_j$ be $\sigma$-finite measures on $(X_j, \mathcal M_j)$ such that $v_j \ll \mu_j$. Prove that $v_1 \times v_2 \ll \mu_1 \times \mu_2$ and $${d(v_1 \times v_2) \over d(\mu_1 \times \mu_2)}(x_1, x_2) = {dv_1 \over d\mu_1}(x_1){dv_2 \over d\mu_2}(x_2)$$
Hint: Since $\nu_j<<\mu_j$ and the measures are $\sigma$-finite, you can apply the Lebesgue-Radon-Nikodym theorem. Then apply Fubini-Tonelli's theorems to get the results (you'll need to use Fubini-Tonelli more than once).