I am in a Stochastic calc class right now and we are defining the Ito integral. Our definition of a simple process is:
A process $X \in L_2$ is simple if there exists a countable partition $\Pi$ st. $X_t(\omega) = X_{t_j}(\omega)$ for all $t \in [t_j,t_{j+1})$ for all $\omega \in \Omega$.
There is a note immediately after that says:
It is important to note that we assume the partition does not depend on $\omega$. Thus not every piece-wise constant process is a simple process. Give an example of such.
I can not seem to think of an example, any guidance would be appreciated.