Could anyone offere an example of two stochastic processes that are modifications of each other, i.e.
$P(A_t)=1\qquad \forall t\in [0,T]$,$\qquad$ with $A_t:=\{\omega\in\Omega:X_t=Y_t\}$,
and such that
$P(\omega\in\Omega: \mu(I_{\omega})>0)>0$, with $I_{\omega}:=\{t\in [0,T] : \omega \in A^c_t \}$, and where $\mu$ is the Lebesgue measure on $[0,T]$?
Let me give you a bit of context. I am trying to understand how different could be, in principle, the trajectories of two versions of the same process. So far, all I could find is the typical textbook example of a modification that is not indistinguishable, which is simple but not satisfactory as the trajectories only differ at one time.
My intuition (which could be wrong of course) is that a modification like the one above should exist, but I struggle to find a formal argument to prove it.
Thanks in advance!
As $\mathbb{P}(A_t^c)=0$ for each $t \geq 0$, we have by Tonelli's theorem
$$\mathbb{E} \left( \int_0^{T} 1_{A_t^c} \, dt \right) = \int_0^{T} \mathbb{P}(A_t^c)=0,$$
and so
$$\int_0^{T} 1_{A_t^c} \, dt = 0 \quad \text{a.s.}$$
Since $1_{A_t^c}(\omega) = 1_{I_{\omega}}(t)$ this implies that
$$\int_0^T 1_{I_{\omega}}(t) \, dt=0 \quad \text{a.s.},$$
i.e. $\mu(I_{\omega})=0$ almost surely. This means that there doesn't exist a modification with "very different" sample paths.