Product measure and absolute continuity.

Question

Suppose that $(\Omega_1,\mathcal{F}_1,\mu_1)$ and $(\Omega_2,\mathcal{F}_2,\mu_2)$ are measure spaces. Suppose that on $\mathcal{F}_1$ is defined another measure $\mu_0$ such that $\mu_0\ll\mu_1$, i.e. $\mu_0$ is absolutely continuous with respect to $\mu_1$. Let $H\in \mathcal{F}_1\otimes \mathcal{F}_2$ be such that $$ (\mu_1\otimes\mu_2)(H) = 0. $$ Can I conclude that $$ (\mu_0\otimes\mu_2)(H) = 0 ? $$ My guess is yes. In fact $\mu_0\ll\mu_1$ implies that if $A\in\mathcal{F}_1$ is such that $\mu_1(A) = 0$ then $\mu_0(A)=0$. Nevertheless, I am struggling to prove the result for a generic $H$ in $ \mathcal{F}_1\otimes \mathcal{F}_2$ (with $\mu_1\otimes\mu_2$ zero measure ).

Answer 1

Taken from some lecture notes that I am writing.