One step in the proof of Tonelli-Fubini Theorem
$Q$ is a probability measure on $(F,\mathcal{F})$.
Suppose $C\in\mathcal{E}\bigotimes\mathcal{F}$ and $C(x)=\{y:(x,y)\in C\}$
Let $\mathcal{H}=\left\{ C\in\mathcal{E}\bigotimes\mathcal{F}:\: x\rightarrow Q[C(x)]\textrm{ is }\mathcal{E}\textrm{-measurable}\right\} $ .
Show that $\mathcal{H}$ is closed under increasing limit and differences.
I can prove it is closed under the increasing limit because $C_{n}(x)\uparrow C(x)\Rightarrow Q[C_{n}(x)]\uparrow Q[C(x)]$.
However, I don't know how to show the difference because
(1) The difference of two measurable functions might not be measurable.
(2) $Q[C_{1}(x)\backslash C_{2}(x)]\neq Q[C_{1}(x)]-Q[C_{2}(x)]$ .
Thank you!