Let $\mu_T(\omega,ds)=dA_s(\omega)$ be a Lebesgue- Stieljtes measure on $[0,T]$ induced by the increasing, right-continuous adapted process $s \mapsto A_s(\omega)$, i.e. $\mu_T(\omega, [s,t)) = A_t(\omega)- A_s(\omega)$ for all $s \le t \le T$. Let $m(s,\omega)$ be a $\lambda_T \otimes P$ integrable process.
In this case, is the process $(t,\omega) \mapsto \int_0^t m(s,\cdot) dA_s$ is adapted, i.e. $\mathscr{F}_t$- measurable even when $m$ is not progressively measurable?
For the case where $m$ is progressively measurable, we can use a monotone class argument.
But I cannot really think of a proof without progressively measurability. Is this statement true in general?