# Example of non measurable stochastic process

measurable-functionsstochastic-analysisstochastic-calculusstochastic-processes

I know that a Stochastic process $$X=(\Omega,\mathcal{F},\{\mathcal{F}_t\}_{t\in T},\{X_t\}_{t\in T}),P)$$ is said to be measurable if the map:
$$G:(T\times\Omega,\mathcal{B}(T)\times\mathcal{F})\to(E,\mathcal{B}(E))$$
such that $$(t,\omega)\mapsto X_t(w)$$ is a measurable map. ($$\mathcal{B}(E)$$ denotes borelian of $$E$$ )

I'm looking for an exampke of a non measurable stochastic process but I still struggle on finding it. Can someone give me an hint?

Take $$\mathcal{F}=\{\emptyset,\Omega\}$$. Then any non-deterministic process is non measurable. For a more interesting example, consider a a family of i.i.d. random variables $$(X_t:t\in[0,1])$$ s.t. $$\mathsf{E}X_0=0$$ and $$\mathsf{E}X_0^2=1$$. It is easy to see that if $$\{X_t\}$$ is measurable, then for any $$0\le a, $$\mathsf{E}\left[\int_{[a,b]}X_t\, dt\right]^2=\int_{[a,b]^2}\mathsf{E}[X_sX_t]\, dsdt=0$$ (by Fubini), i.e., $$X_t=0$$ for almost all $$\omega\in\Omega$$ and $$t\in[0,1]$$. Thus, $$\mathsf{E}\left[\int_0^1X_t^2\,dt\right]=0\ne 1=\int_0^1 \mathsf{E}X_t^2\, dt.$$