While reading a paper, I've witnessed the following application of the mean value theorem
$$\frac{EG((x-\epsilon)X_{\tau_\epsilon})-EG(xX_{\tau_\epsilon})}{-\epsilon}=E[G'(\xi_\epsilon)X_{\tau_\epsilon}]$$ for some $\xi_\epsilon \in [(x-\epsilon)X_{\tau_\epsilon},xX_{\tau_\epsilon}],$ by the mean value theorem. Here, $X$ is a geometric brownian motion, $\epsilon>0$, $\tau_\epsilon$ is an optimal stopping time for $V(t,x-\epsilon)=\inf_{0\le \tau \le T-t}EG(xX_\tau)$, where the infimum is taken over the stopping times, and $G$ is a $C^1$ function. However, I don't know exactly which mean value theorem it is referring to, since for the expectation on the right hand side to make sense, we need the random variable $\xi_\epsilon$ to be measurable, but the mean value theorem from calculus does not guarantee this. Is there a well known theorem about this, or is there a way to show that the random variable $\xi_\epsilon$ is measurable?