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)

Formulas involving the existential and universal quantifiers $\exists$ and $\forall$ can be rephrased in terms of unions and intersections.  In this case, you can write:
$$P\left[\exists n\geq 1: X_{n}=0\right] = P\left[\bigcup_{n\geq 1}\{\omega: X_{n}(\omega)=0\}\right]$$
