Let $X$ be a stochastic process and $T$ a stopping time. Then one forms the random variable $X_T$.
I have a quite vague question: apparently $X_T$ will be quite different if we replace $X$ by one of its modifications, does this matter? Is it the case $X_T$ has to be defined only up to indistinguishable $X$ but not modification? Or are there any special conditions (e.g. one can demand modifications that preserve right continuous) under which modification is possible?