Measurability of a function related to Skorohod space Let $(E,r)$ be a metric space, $D_E[0,\infty)$ be the Skorohod space on $[0,\infty)$ takes value in $E$. Consider the function
$$(D_E[0,\infty)\times[0,\infty),\mathcal B(D_E[0,\infty)\times \mathcal B([0,\infty))\to (E,\mathcal B(E))$$
$$(x,t)\mapsto x_t$$
I am wondering whether this map is measurable? To fix $t$, $x\mapsto x_t$ is measurable by the property of Skorohod space. But for the function above I can't prove whether it is measurable. Any help, please. Thanks very much!

You can exploit the right continuity of the sample paths to get joint measurability. For $n\geq 1$ define $\Phi_n(x,t)=x_{\lceil nt \rceil/n}$.
This map $\Phi_n$ is jointly measurable, and $\Phi_n(x,t)\to x_t$ as $n\to\infty$.  
