Let $a < b$. Let $X_t$ be a right continous stochastic process that is $\mathcal{F_t}$ adapated.
For any finite subset $F$ of interval $I \subseteq [0,\infty)$, define the $U_{F}(a,b,X)$ to be the number of upcrossings between levels $(a,b)$ in X’s with indices from F. Define
$U_I(a,b,X) =\sup\{U_F(a,b,X): F \text{ is a finite subset of } I\}$.
I want to prove that $U_I$ is a measurable function. I know $U_F(a,b,X)$ is measurable for every finite subset $F$. Any hints on proving measurability of $U_I(a,b,X)$ would be helpful.