Consider the stochastic process $\{X(t)\}_{t\in T}$ and certain specific operations $g$ and $f$. If $T$ is of finite dimension, i.e. $T=\{t_1,\ldots,t_k\}$, with $t_i\in 0,\infty)$ then I have been able to proof that the two operation coincide $$g(X(t))=f(X(t)), \ t\in T.$$
I am now looking to extend this result to $T=[0,\infty)$.
Does anyone know how to go about this? I am also looking for good references on this topic.