I have a process $(X,Y)\in D([0,T],\mathbb{R}^2)$ where $D([0,T],\mathbb{R}^2)$ is the set of cadlag functions with Skorohod metric. Let $A=\{\sup_{t\in[0,T]}|X(t)-\int_0^tX(s)Y(s)ds|>\epsilon\}$ where $\epsilon$ is a positive constant.
I need to prove that the characteristic function $\mathbb{I}_A$ is continuous (but I'm not sure that this is true).
If I can prove that the function $t\to\sup_{t\in[0,T]}|X(t)-\int_0^tX(s)Y(s)ds|$ is continuous I would have that $A$ is open and than $\mathbb{I}_A$ would be continuous.
Does anybody know a reference to a proof of that fact, or know how to prove it?
Thanks for any help and sorry if my english wasn't correct.