# On correspondence of random function and stochastic processes

By reading Billingsly book on convergence of probability measures I was led to believe the following,

Given a random function $X:\Omega \rightarrow \mathcal{X}$ we obtain the stochastic processes $(X_{t})_{t \in T}$ by considering $X_{t}=\pi_{t}\circ X$ where $\pi_{t} :\mathcal{X}\rightarrow \mathbb{R}$ is the point evaluation functional for some function space $\mathcal{X}$ (where $X$ takes its values).

Can we obtain any stochastic process of this way if we choose the right $\mathcal{X}$?

• Sure, try $\mathcal X=\mathbb R^T$. – Did Apr 14 '17 at 6:12
• @Did ah I tought that it was only continuous function in that space. Thanks. – user1 Apr 14 '17 at 6:19

Just take $\mathcal{X}=R^{T}$.