Almost sure equality only makes sense for random variables on the same probability space
Note that the expression
\begin{equation}
\mathbb{P}(X \neq Y)
= \mathbb{P} \{ \omega \in \Omega : X(\omega) \neq Y(\omega) \}
\end{equation}
only makes sense if $X$ and $Y$ are defined on a common probability space $P = (\Omega, \mathcal{F}, \mathbb{P})$.
