About the notation in probability theory I am reading about recurrence random walks and I see the following expression:
$$\mathbb{P}(\exists n\geq 1,X_n = 0) = 1$$
So, I wonder: what is this precisely? I know, for instance,
$$\mathbb{P}(X_n = 0) = \mathbb{P}(\{\omega:X_n(\omega) = 0\})$$
rigorously. But I couldn't make sense of the first expression in the same language I gave in the second expression. In some sources, they say that this implies that $X_n$ visiting the zero position infinitely often (implies or same thing?). This is probably an issue of notation but if someone explains, it would be truly helpful. Thanks!
(see e.g. Proposition 3 from this pdf)

Let $R$ denote a property each $\omega$ in $\Omega$ may have or not, and $A=\{\omega\in\Omega\mid R(\omega)\}$, in your case $R(\omega)$ is: $\exists n\geqslant1,X_n(\omega)=0$.
Then, $P(R)$ is simply $P(A)$, that is,
$$
P(\exists n\geqslant1,X_n=0)=P(\{\omega\in\Omega\mid \exists n\geqslant1,X_n(\omega)=0\}).
$$
By the way, this is exactly for the same reason that
$$
P(X_n=0)=P(\{\omega\in\Omega\mid X_n(\omega)=0\}).
$$
