Measurablity of functions defined over sections of product measures I have to solve the following exercise but I am unable to proceed. Could you please give me some hints to how to solve it?
Let $(\Omega_1, \mathcal{F}_1)$ and $(\Omega_2, \mathcal{F}_2)$ be measurable spaces. Take $\mathcal{F}_1\otimes \mathcal{F}_2$ to be the product $\sigma$-field for $\Omega_1\times\Omega_2$. Also for any $B\in \mathcal{F}_1\otimes \mathcal{F}_2$, take $B_{\omega_1}:=\{\omega_2 : (\omega_1, \omega_2) \in B\}$, the $\omega_1$-section of $B$. Show that for a finite measure $\mu_2$ for $(\Omega_2, \mathcal{F}_2)$, $f(\omega_1):=\mu_2(B_{\omega_1})$ is a measurable function from $\Omega_1$ to $\mathbb{R}$.
The problem is too complex for me to comprehend it in a reasonable way. I understand that it is enough to show that $f^{-1}((-\infty,k])\in \mathcal{F}_1$, but this does not seem any simpler. 

Use the monotone class theorem. That is let $\mathcal{C}$ be the set of all $B\in\mathcal{F}_1\otimes \mathcal{F}_2$ for which $f$ is measurable. Then if $B=A\times C$, $\mu_2(B_{\omega_1})=1_A(\omega_1)\times \mu_2(C)$, which is measurable so that $f$ is measurable on the finite disjoint unions of rectangles (an algebra contained in $\mathcal{C}$). It remains to show that $\mathcal{C}$ is closed under countable increasing unions and countable decreasing intersections.
