Let $(E,\mathscr E)$ be a measurable space and $Q:E\times \mathscr E\to\Bbb [-1,1]$ be a signed bounded kernel, i.e. $Q_x(\cdot)$ is a finite measure on $(E,\mathscr E)$ for any $x\in E$ and $x\mapsto Q_x(A)$ is a measurable function for any set $A\in \mathscr E$.
For any fixed $x$, let the measure $Q^+_x$ be a positive part of the signed measure $Q_x$ as in Hahn-Jordan decomposition. Is it true that $Q^+$ is a kernel, i.e. is the function $x\mapsto Q_x^+(A)$ measurable for any $A\in \mathscr E$? It clearly holds if $Q$ is an integral kernel, i.e. $$ Q(x,\mathrm dy) = q(x,y)\mu(\mathrm dy) $$ where $\mu$ is a finite measure on $(E,\mathscr E)$ and $q:E\times E\to \mathbb R$ is a jointly measurable function, but I am interested in the general case.
Any hints oh how to prove that the measurability holds in general, or on how to derive a counter-example.
This question I asked on MSE, but in three weeks no ideas were mentioned there. Neither helped the bounty which has recently finished. Due to this reason, I hope, it's ok to post the question here.