Let $(\Omega_1,\mathcal{A}_1,\mu_1)$, $(\Omega_2,\mathcal{A}_2,\mu_2)$ be two finite measure spaces and denote by $(\Omega_1 \times \Omega_2,\mathcal{A}_1 \otimes \mathcal{A}_2,\mu_1 \otimes \mu_2)$ the product measure space. I would like to know if the following statement is true:
Let $f \in L_1(\mu_1 \otimes \mu_2)$ and $g:\Omega_1 \times \Omega_2 \rightarrow \mathbb{R}$ (not necassarily measurable!). If $f(x,\cdot) = g(x,\cdot)$ $\mu_2$-a.e. for $\mu_1$-a.e. $x \in \Omega_1$, then $g$ is $\mu_1 \otimes \mu_2$-measurable and the identity $f = g$ holds $\mu_1 \otimes \mu_2$-almost everywhere.
If this statement does not hold so generally, does it perhaps hold on more specific measure spaces, e.g., on open bounded subsets of $\mathbb{R}^n$?
Could eventually someone help me out here?
Thank you in advance!