# Example of indistinguishable process that are not s.t. $X_t(\omega )=Y_t(\omega )$ for all $t$ and all $\omega$.

Let $$(\Omega ,\mathcal F,\mathbb P)=([0,1],\mathcal B([0,1],m)$$ where $$m$$ is the Lebesgue measure. Let $$X_t=0$$ for all $$t$$ and $$Y_t(\omega )=\mathbb 1_{t=\omega }$$.

Even if $$X_t(\omega )=Y_t(\omega )$$ for all $$t$$ and all $$\omega \neq t$$ (i.e. for all $$t$$, except at one point, they are the same processes), we have that $$\mathbb P(\forall t\geq 0, X_t=Y_t)=0.$$ So it looks that to be indistinguishable, we should have $$X_t(\omega )=Y_t(\omega )$$ for all $$t$$ and all $$\omega$$. But, if this would be the case, I guess that we would define indstinguishable processes as $$X_t(\omega )=Y_t(\omega )$$ for all $$t$$ and all $$\omega$$, and not as $$\mathbb P(\forall t, X_t=Y_t)=1$$. So, are there indistinguishable processes $$(X_t)$$ and $$(Y_t)$$ s.t. $$X_t(\omega )= Y_t(\omega )$$ for all $$t$$ and all $$\omega$$ doesn't hold ?

Two processes $$(X_t)_{t \geq 0}$$ and $$(Y_t)_{t \geq 0}$$ are a modification of each other if

$$\mathbb{P}(X_t=Y_t)=1 \quad \text{for all t \geq 0},$$

i.e. for each $$t \geq 0$$ there exists an exceptional null set $$N=N(t)$$ such that $$X_t(\omega) = Y_t(\omega) \quad \text{for all \omega \in \Omega \backslash N(t)}.$$

If the processes $$(X_t)_{t \geq 0}$$ and $$(Y_t)_{t \geq 0}$$ are indistinguishable, then the exceptional null set can be chosen independently from $$t$$, i.e. there exists a null set $$N$$ such that

$$X_t(\omega) = Y_t(\omega) \quad \text{for all t \geq 0, \omega \in \Omega \setminus N}.$$

Equivalently, $$\mathbb{P}(\forall t \geq 0: X_t=Y_t)=1$$. This means that the processes can differ on a null set. Note that null sets might be quite "large", depending on the measure which we consider.

Take, for instance, $$\Omega = [0,1]$$ with the Dirac measure $$\mathbb{P}:=\delta_0$$, and define

$$X_t(\omega) := \sin(\omega t) \qquad Y_t(\omega)=0.$$

We have $$X_t(0)=0=Y_t(0)$$ and so

$$\mathbb{P}(\forall t \geq 0: X_t=Y_t) \geq \delta_0(\{0\})=1.$$

Consequently, $$(X_t)_{t \geq 0}$$ and $$(Y_t)_{t \geq 0}$$ are indistinguishable although they are (from the "natural" point of view) quite different processes. In particular, $$X_t(\omega) \neq Y_t(\omega)$$ for "many" $$t$$ and $$\omega$$.

In your example take $$Y_t(\omega)=1_{t}(\omega)$$ for $$\omega\in N$$ and $$0$$, otherwise, where $$N\ne\emptyset$$ but $$m(N)=0$$.