All the time, I see people working with a given Brownian motion $(B_t)_{t\ge 0}$ on a fixed probability space $(\Omega,\mathcal A,\operatorname P)$ and suddenly exploiting its almost sure properties as if they were sure properties.
Yeah, I know we can find a $\operatorname P$-null set $N\subseteq\Omega$ such that these properties hold surely on $\tilde\Omega:=\Omega\setminus N$ and hence coud replace $(\Omega,\mathcal A,\operatorname P)$ by $(\tilde\Omega,\left.\mathcal A\right|_{\tilde\Omega},\left.\operatorname P\right|_{\tilde\Omega})$. As a second option, we can modify $X_t$ on $N$ for any $t\ge 0$ such that these properties hold surely on $\Omega$.
However, either we need to modify the probability space or the Brownian motion. But that's a weird and unusual practice in mathematics.
If we can't prove the desired result, then we just modify our objects such that we can prove them. Carry the world as you like it.
Shouldn't we better state (in a theorem) that there exists $B$ and $(\Omega,\mathcal A,\operatorname P)$ such that ... instead of fixing $B$ and $(\Omega,\mathcal A,\operatorname P)$ (at a global space)?
And that's just an example. In probability theory, we do things like that all the time. Why is it legitimate?