Given a process $X_t$ and another process $T_t$ which is increasing, what conditions should we require such that the process $X_t$ is adapted to the time change $T_t$, that is such that $X_t$ is constant in the interval $[T_{t-},T_t]$?
I know that if $T_t$ is continuous the above is satisfied. However, what happens if $T_t$ moves by jumps and is constant between jump times (is a subordinator)?