I would be very gratefull if someone could help me with my question below. Intuitivly I can see that it is correct but I am unsure of how to prove it.
Let T be a stopping time in $\mathcal{F}_t$ for $t \geq 0$. $Y$ is a random variable that is $\mathcal{F}_T$ measurable.
The process $(X_t)_{t\geq0}$ is defined by $(X_t)=0$ for $0\leq t\leq T$ and $(X_t)=Y$ when $t \gt T$.
I want to show that $(X_t)_{t\geq0}$ is a predictable process. I can see that the process is left continous and therefore should also be predictable but i am unsure how to prove this. Would be very thankfull for any advise on where to start.